Methods of Patel Loadflow Computation for Electrical Power System

ABSTRACT

Propounding statement of Patel Numerical Method (PNM) for solution of simultaneous algebraic equations, both linear and non-linear, is presented. A new class of Patel Loadflow Methods are invented. These invented Patel Loadflow Methods are Patel Loadflow-1 (PL-1) PL-2, Patel Super Decoupled Loadflow-1 (PSDL-YY1), PSDL-YY2, C-matrix based Patel Loadflow-1 (CPL-1), CPL-2, Sparse Z-matrix based Patel Loadflow {SZPL or S[C] −1 PL (SCIPL)}, and Gauss-Seidel-Patel Loadflow (GSPL) that can also be developed into Decoupled GSPL-method.

FIELD OF THE INVENTION

The present invention relates to a method of loadflow computation inpower flow control and voltage control for an electrical power system.

BACKGROUND OF THE INVENTION

The present invention relates to power-flow/voltage control inutility/industrial power networks of the types including many powerplants/generators interconnected through transmission/distribution linesto other loads and motors. Each of these components of the power networkis protected against unhealthy or alternatively faulty, over/undervoltage, and/or over loaded damaging operating conditions. Such aprotection is automatic and operates without the consent of powernetwork operator, and takes an unhealthy component out of service bydisconnecting it from the network. The time domain of operation of theprotection is of the order of milliseconds.

The purpose of a utility/industrial power network is to meet theelectricity demands of its various consumers 24-hours a day, 7-days aweek while maintaining the quality of electricity supply. The quality ofelectricity supply means the consumer demands be met at specifiedvoltage and frequency levels without over loaded, under/over voltageoperation of any of the power network components. The operation of apower network is different at different times due to changing consumerdemands and development of any faulty/contingency situation. In otherwords healthy operating power network is constantly subjected to smalland large disturbances. These disturbances could be consumer/operatorinitiated, or initiated by overload and under/over voltage alleviatingfunctions collectively referred to as security control functions andvarious optimization functions such as economic operation andminimization of losses, or caused by a fault/contingency incident.

For example, a power network is operating healthy and meeting qualityelectricity needs of its consumers. A fault occurs on a line or atransformer or a generator which faulty component gets isolated from therest of the healthy network by virtue of the automatic operation of itsprotection. Such a disturbance would cause a change in the pattern ofpower flows in the network, which can cause over loading of one or moreof the other components and/or over/under voltage at one or more nodesin the rest of the network. This in turn can isolate one or more othercomponents out of service by virtue of the operation of associatedprotection, which disturbance can trigger chain reaction disintegratingthe power network.

Therefore, the most basic and integral part of all other functionsincluding optimizations in power network operation and control issecurity control. Security control means controlling power flows so thatno component of the network is over loaded and controlling voltages suchthat there is no over voltage or under voltage at any of the nodes inthe network following a disturbance small or large. As is well known,controlling electric power flows include both controlling real powerflows which is given in MWs, and controlling reactive power flows whichis given in MVARs. Security control functions or alternatively overloadsalleviation and over/under voltage alleviation functions can be realizedthrough one or combination of more controls in the network. Theseinvolve control of power flow over tie line connecting other utilitynetwork, turbine steam/water/gas input control to control real powergenerated by each generator, load shedding function curtails loaddemands of consumers, excitation controls reactive power generated byindividual generator which essentially controls generator terminalvoltage, transformer taps control connected node voltage, switchingin/out in capacitor/reactor banks controls reactive power at theconnected node.

Control of an electrical power system involving power-flow control andvoltage control commonly is performed according to a process shown inFIG. 5, which is a method of forming/defining and solving a loadflowcomputation model of a power network to affect control of voltages andpower flows in a power system comprising the steps of:

-   Step-10: obtaining on-line/simulated data of open/close status of    all switches and circuit breakers in the power network, and reading    data of operating limits of components of the power network    including maximum power carrying capability limits of transmission    lines, transformers, and PV-node, a generator-node where    Real-Power-P and Voltage-Magnitude-V are    given/assigned/specified/set, maximum and minimum reactive power    generation capability limits of generators, and transformers tap    position limits, or stated alternatively in a single statement as    reading operating limits of components of the power network,-   Step-20: obtaining on-line readings of given/assigned/specified/set    Real-Power-P and Reactive-Power-Q at PQ-nodes, Real-Power-P and    voltage-magnitude-V at PV-nodes, voltage magnitude and angle at a    reference/slack node, and transformer turns ratios, wherein said    on-line readings are the controlled variables/parameters,-   Step-30: performing loadflow computation to calculate, depending on    loadflow computation model used, complex voltages or their real and    imaginary components or voltage magnitude corrections and voltage    angle corrections at nodes of the power network providing for    calculation of power flow through different components of the power    network, and to calculate reactive power generation and transformer    tap-position indications,-   Step-40: evaluating the results of Loadflow computation of step-30    for any over loaded power network components like transmission lines    and transformers, and over/under voltages at different nodes in the    power system,-   Step-50: if the system state is acceptable implying no over loaded    transmission lines and transformers and no over/under voltages, the    process branches to step-70, and if otherwise, then to step-60,-   Step-60: correcting one or more controlled variables/parameters set    in step-20 or at later set by the previous process cycle step-60 and    returns to step-30,-   Step-70: affecting a change in power flow through components of the    power network and voltage magnitudes and angles at the nodes of the    power network by actually implementing the finally obtained values    of controlled variables/parameters after evaluating step finds a    good power system or stated alternatively as the power network    without any overloaded components and under/over voltages, which    finally obtained controlled variables/parameters however are stored    for acting upon fast in case a simulated event actually occurs or    stated alternatively as actually implementing the corrected    controlled variables/parameters to obtain secure/correct/acceptable    operation of power system.

Overload and under/over voltage alleviation functions produce changes incontrolled variables/parameters in step-60 of FIG. 5. In other wordscontrolled variables/parameters are assigned or changed to the newvalues in step-60. This correction in controlled variables/parameterscould be even optimized in case of simulation of all possible imaginabledisturbances including outage of a line and loss of generation forcorrective action stored and made readily available for acting upon incase the simulated disturbance actually occurs in the power network. Infact simulation of all possible imaginable disturbances is the modernpractice because corrective actions need be taken before the operationof individual protection of the power network components.

It is obvious that loadflow computation consequently is performed manytimes in real-time operation and control environment and, therefore,efficient and high-speed loadflow computation is necessary to providecorrective control in the changing power system conditions including anoutage or failure of any of the power network components. Moreover, theloadflow computation must be highly reliable to yield converged solutionunder a wide range of system operating conditions and networkparameters. Failure to yield converged loadflow solution creates blindspot as to what exactly could be happening in the network leading topotentially damaging operational and control decisions/actions incapital-intensive power utilities.

The power system control process shown in FIG. 5 is very general andelaborate. It includes control of power-flows through network componentsand voltage control at network nodes. However, the control of voltagemagnitude at connected nodes within reactive power generationcapabilities of electrical machines including generators, synchronousmotors, and capacitor/inductor banks, and within operating ranges oftransformer taps is normally integral part of loadflow computation asdescribed in “LTC Transformers and MVAR violations in the Fast DecoupledLoad Flow, IEEE Trans., PAS-101, No. 9, PP. 3328-3332, September 1982.”If under/over voltage still exists in the results of loadflowcomputation, other control actions, manual or automatic, may be taken instep-60 in the above and in FIG. 5. For example, under voltage can bealleviated by shedding some of the load connected.

The prior art and present invention are described using the followingsymbols and terms:

Y_(pq)=G_(pq)+jB_(pq): (p−q) th element of nodal admittance matrixwithout shuntsY_(pp)=G_(pp)+jB_(pp): p-th diagonal element of nodal admittance matrixwithout shuntsy_(p)=g_(p)+jb_(p): total shunt admittance at any node-pV_(p)=e_(p)+jf_(p)=V_(p)∠θ_(p): complex voltage of any node-pV_(s)=e_(s)+jf_(s)=V_(s)∠θ_(s): complex slack-node voltageΔθ_(p),ΔV_(p): voltage angle, magnitude correctionsΔf_(p),Δe_(p): imaginary, real part of complex voltage correctionsS_(p)=P_(p)+jQ_(p): net nodal injected power, calculatedΔP_(p)+jΔQ_(p): nodal power residue or mismatchRP_(p)+jRQ_(p): modified nodal power residue or mismatchRI_(p)+jII_(p): net nodal injected current, calculatedΔRI_(p)+jΔII_(p): nodal injected current residue or mismatchRRI_(p)+jRII_(p): modified nodal current residue or mismatchSSH_(p)=PSH_(p)+jQSH_(p): net nodal injected power, scheduled/specifiedC_(p)=1∠Φ_(p)=Cos Φ_(p)+jSin Φ_(p): Unitary rotation/transformationm: number of PQ-nodesk: number of PV-nodesn=m+k+1: total number of nodesq>p: node-q is connected to node-p excluding the case of q=p[ ]: indicates enclosed variable symbol to be a vector or matrixLRA: Limiting Rotation Angle, −48° for invented modelsPQ-node: load-node, where, Real-Power-P and Reactive-Power-Q arespecifiedPV-node: generator-node, where, Real-Power-P and Voltage-Magnitude-V arespecifiedV_(s)≈V_(B)≈V_(N): Slack-node voltage magnitude, Base value, and Nominalvalue of voltage magnitude are very closely similar, and therefore, theycan be used interchangeably. However, in the following development onlyV_(s) will be used. Particularly, in the treatment of loadflow problemwith distributed slack-node, there is no specific slack-node and VB orV_(N) can be used.

-   Loadflow Computation: Each node in a power network is associated    with four electrical quantities, which are voltage magnitude,    voltage angle, real power, and reactive power. The loadflow    computation involves calculation/determination of two unknown    electrical quantities for other two    given/specified/scheduled/set/known electrical quantities for each    node. In other words the loadflow computation involves determination    of unknown quantities in dependence on the    given/specified/scheduled/set/known electrical quantities.-   Loadflow Model: a set of equations describing the physical power    network and its operation for the purpose of loadflow computation.    The term ‘loadflow model’ can be alternatively referred to as ‘model    of the power network for loadflow computation’. The process of    writing Mathematical equations that describe physical power network    and its operation is called Mathematical Modeling. If the equations    do not describe/represent the power network and its operation    accurately the model is inaccurate, and the iterative loadflow    computation method could be slow and unreliable in yielding    converged loadflow computation. There could be variety of Loadflow    Models depending on organization of set of equations describing the    physical power network and its operation, including Decoupled    Loadflow Models, Super Decoupled Loadflow Models, Fast Super    Decoupled Loadflow (FSDL) Model, and Super Super Decoupled Loadflow    (SSDL) Model.-   Loadflow Method: sequence of steps used to solve a set of equations    describing the physical power network and its operation for the    purpose of loadflow computation is called Loadflow Method, which    term can alternatively be referred to as ‘loadflow computation    method’ or ‘method of loadflow computation’. One word for a set of    equations describing the physical power network and its operation    is: Model. In other words, sequence of steps used to solve a    Loadflow Model is a Loadflow Method. The loadflow method involves    definition/formation of a loadflow model and its solution. There    could be variety of Loadflow Methods depending on a loadflow model    and iterative scheme used to solve the model including Decoupled    Loadflow Methods, Super Decoupled Loadflow Methods, Fast Super    Decoupled Loadflow (FSDL) Method, and Super Super Decoupled Loadflow    (SSDL) Method. All decoupled loadflow methods described in this    application use either successive (1θ, 1V) iteration scheme or    simultaneous (1V, 1θ) iteration scheme, defined in the following.

Prior art method of loadflow computation of the kind carried out asstep-30 in FIG. 7, include a class of methods known as decoupledloadflow. This class of methods consists of decouled loadflow and superdecoupled loadflow methods including Super Super Decoupled Loadflowmethod all formulated involving Power Mismatch computation and polarcoordinates. Prior-art Loadflow Computation Methods are described indetails in the documents of Research publications and granted patentscited in Information Disclosure Statement (IDS) by this inventor.Therefore, prior art methods will not be described here.

SUMMARY OF THE INVENTION

It is a primary object of the present invention to improve convergenceand efficiency of the prior art Super Super Decoupled Loadflowcomputation method under wide range of system operating conditions andnetwork parameters for use in power flow control and voltage control inthe power system. A further object of the invention is to reducecomputer storage/memory or calculating volume requirements.

The above and other objects are achieved, according to the presentinventions, Patel Loadflow (PL-1 & PL-2), Patel Super Decoupled Loadflow(PSDL-YY1 & PSDL-YY2), Y matrix—Patel Loadflow (YPL-1 & YPL2), Sparse Zor C⁻¹ matrix—Patel Loadflow (SZPL or SCIPL), Guass-Seidel-PatelLoadflow (GSPL) Methods and their many variants, for loadflowcalculation for Electrical Power System. In context of voltage control,one of the inventive system of PSDL-YY2 and others listed in the abovemethods of loadflow computation is used for Electrical Power systemconsisting of plurality of electromechanical rotating machines,transformers and electrical loads connected in a network, each machinehaving a reactive power characteristic and an excitation element whichis controllable for adjusting the reactive power generated or absorbedby the machine, and some of the transformers each having a tap changingelement, which is controllable for adjusting turns ratio oralternatively terminal voltage of the transformer, said systemcomprising:

-   -   means defining and solving one of the loadflow models of the        power network listed in the above for providing an indication of        the quantity of reactive power to be supplied by each generator        including the reference/slack node generator, and for providing        an indication of turns ratio of each tap-changing transformer in        dependence on the obtained-online or given/specified/set/known        controlled network variables/parameters, and physical limits of        operation of the network components,    -   machine control means connected to the said means defining and        solving loadflow model and to the excitation elements of the        rotating machines for controlling the operation of the        excitation elements of machines to produce or absorb the amount        of reactive power indicated by said means defining and solving        loadflow model in dependence on the set of obtained-online or        given/specified/set controlled network variables/parameters, and        physical limits of excitation elements,    -   transformer tap position control means connected to the said        means defining and solving loadflow model and to the tap        changing elements of the controllable transformers for        controlling the operation of the tap changing elements to adjust        the turns ratios of transformers indicated by the said means        defining and solving loadflow model in dependence on the set of        obtained-online or given/specified/set controlled network        variables/parameters, and operating limits of the tap-changing        elements.

The method and system of voltage control according to the preferredembodiment of the present invention provide voltage control for thenodes connected to PV-node generators and tap changing transformers fora network in which real power assignments have already been fixed. Thesaid voltage control is realized by controlling reactive powergeneration and transformer tap positions.

One of the inventive methods of defining and solving loadflowcomputation models PL-1, PL-2, PSDL-YY1, PSDL-YY2, YPL-1, YPL-2, SZPL orGSPL can be used for voltage control in Electrical power System. Forthis purpose real and reactive power assignments or settings atPQ-nodes, real power and voltage magnitude assignments or settings atPV-nodes and transformer turns ratios, open/close status of all circuitbreaker, the reactive capability characteristic or curve for eachmachine, maximum and minimum tap positions limits of tap changingtransformers, operating limits of all other network components, and theimpedance or admittance of all lines are supplied. A decoupled loadflowsystem of equations {(28) and (29)} or {(30) and (31)} is solved by aniterative process until convergence. During this solution the quantitieswhich can vary are the real and reactive power at the reference/slacknode, the reactive power set points for each PV-node generator, thetransformer transformation ratios, and voltages on all PQ-nodes nodes,all being held within the specified ranges. When the iterative processconverges to a solution, indications of reactive power generation atPV-nodes and transformer turns-ratios or tap-settings are provided.Based on the known reactive power capability characteristics of eachPV-node generator, the determined reactive power values are used toadjust the excitation current to each generator to establish thereactive power set points. The transformer taps are set in accordancewith the turns ratio indication provided by the system of loadflowcomputation.

For voltage control, system of PSDL-YY2 or others and many variantslisted in the above Methods of Loadflow computation can be employedeither on-line or off-line. In off-line operation, the user can simulateand experiment with various sets of operating conditions and determinereactive power generation and transformer tap settings requirements. Ageneral-purpose computer can implement the entire system. For on-lineoperation, the loadflow computation system is provided with dataidentifying the current real and reactive power assignments andtransformer transformation ratios, the present status of all switchesand circuit breakers in the network and machine characteristic curves insteps-10 and -20 in FIG. 7, and steps 12, 14, 18, 22, 24, 32, 34, and 38in FIG. 8 described below. Based on this information, a model of thesystem based on coefficient matrices of invented loadflow computationsystems provide the values for the corresponding node voltages, reactivepower set points for each machine and the transformation ratio and tapchanger position for each transformer.

The present inventive system of loadflow computation for ElectricalPower System consists of, one of the Patel Super Decoupled Loadflow:YY2-version (PSDL-YY2) or PSDL-X′X′, or others listed in the aboveMethods characterized in that 1) single decoupled coefficient matrixsolution requiring only 50% of memory used by prior art methods, 2) thepresence of transformed values of known/given/specified/scheduled/setquantities in the diagonal elements of the gain matrices [Yf] and [Ye]of the decoupled loadflow sub-problems, and 3) transformation angles arerestricted to maximum of −0° to −90° (say, −48°) to be determinedexperimentally, 4) PV-nodes being active in both RI-f and Thesub-problems, PQ-node to PV-node and PV-node to PQ-node switching issimple to implement, and these inventive loadflow computation stepstogether yield some processing acceleration and consequent efficiencygains, and are each individually inventive, and 5) modified real andimaginary current mismatches at PV-nodes in case of PSDL-YY1, SSDL-YY,HSSDL-YY, ESSDL-YY or their generalized variations PSDL-B′B′, SSDL-B′B′,HSSDL-B′B′, ESSDL-B′B′ are determined asRRI_(p)=(e_(p)ΔP_(p))/[K_(p)(e_(p) ²+f_(p) ²)] andRII_(p)=(−f_(p)ΔP_(p))/[K_(p)(e_(p) ²+f_(p) ²)] in order to keep gainmatrices [Yf] and [Ye] symmetrical. If the value of factor K_(p)=1, thegain matrices [Yf] and [Ye] becomes unsymmetrical in that elements inthe rows corresponding to PV-nodes are defined without transformation orrotation applied, as Yf_(pq)=Ye_(pq)=−B_(pq). It is possible that PatelSuper Decoupled methods can be formulated in polar coordinates by simplyreplacing correction vectors [Δf] and [Δe] in equations (28) and (29)and subsequently followed equations by correction vectors [Δθ] and [ΔV].However, it will not be easy to have single gain matrix model, because[ΔV] for PV-nodes is zero and absent.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 is a flow-chart embodiment of the invented PSDL-YY1 computationmethod.

FIG. 2 is a flow-chart embodiment of the invented PSDL-YY2 computationmethod.

FIG. 3 is a flow-chart embodiment of the invented Y matrix based PatelLoadflow (YPL-1) computation method using complex algebra.

FIG. 4 is a flow-chart embodiment of the invented Y matrix based PatelLoadflow (YPL-2) computation method using complex algebra.

FIG. 5 is a flow-chart embodiment of the invented method of sparse [Z]or [C]⁻¹ based Patel Loadflow (SZPL) or (SCIPL) computation method usingcomplex algebra.

FIG. 6 is a flow-chart embodiment of the invented GSPL computationmethod.

FIG. 7 is a flow-chart of the overall controlling method for anelectrical power system involving loadflow computation as a step whichcan be executed using one of the loadflow computation methods embodiedin FIG. 1, 2, 3, 4, 5 or 6.

FIG. 8 is a flow-chart of the simple special case of voltage controlsystem in overall controlling system of FIG. 5 for an electrical powersystem

FIG. 9 is a one-line diagram of an exemplary 6-node power network havinga reference/slack/swing node, two PV-nodes, and three PQ-nodes

DESCRIPTION OF A PREFERRED EMBODIMENT

A loadflow computation is involved as a step in power flow controland/or voltage control in accordance with FIG. 7 or FIG. 8. A preferredembodiment of the present invention is described with reference to FIG.8 as directed to achieving voltage control.

FIG. 9 is a simplified one-line diagram of an exemplary utility powernetwork to which the present invention may be applied. The fundamentalsof one-line diagrams are described in section 6.11 of the text ELEMENTSOF POWER SYSTEM ANALYSIS, fourth edition, by William D. Stevenson, Jr.,McGrow-Hill Company, 1982. In FIG. 9 each thick vertical line is anetwork node. The nodes are interconnected in a desired manner bytransmission lines and transformers each having its impedance, whichappears in the loadflow models. Two transformers in FIG. 9 are equippedwith tap changers to control their turns ratios in order to controlterminal voltage of node-1 and node-2 where large loads are connected.

Node-6 is a reference/slack-node alternatively referred to as the slackor swing-node, representing the biggest power plant in a power network.Nodes-4 and -5 are PV-nodes where Generators are connected, and nodes-1,-2, and -3 are PQ-nodes where loads are connected. It should be notedthat the nodes-4, -5, and -6 each represents a power plant that containsmany generators in parallel operation. The single generator symbol ateach of the nodes-4, -5, and -6 is equivalent of all generators in eachplant. The power network further includes controllable circuit breakerslocated at each end of the transmission lines and transformers, anddepicted by cross markings in one-line diagram of FIG. 9. The circuitbreakers can be operated or in other words opened or closed manually bythe power system operator or relevant circuit breakers operateautomatically consequent of unhealthy or faulty operating conditions.The operation of one or more circuit breakers modify the configurationof the network. The arrows extending certain nodes represent loads.

A goal of the present invention is to provide a reliable andcomputationally efficient loadflow computation that appears as a step inpower flow control and/or voltage control systems of FIG. 7 and FIG. 8.However, the preferred embodiment of loadflow computation as a step incontrol of terminal node voltages of PV-node generators and tap-changingtransformers is illustrated in the flow diagram of FIG. 8 in whichpresent invention resides in function steps 42 and 44.

Short description of other possible embodiment of the present inventionis also provided herein. The present invention relates to control ofutility/industrial power networks of the types including plurality ofpower plants/generators and one or more motors/loads, and connected toother external utility. In the utility/industrial systems of this type,it is the usual practice to adjust the real and reactive power producedby each generator and each of the other sources including synchronouscondensers and capacitor/inductor banks, in order to optimize the realand reactive power generation assignments of the system. Healthy orsecure operation of the network can be shifted to optimized operationthrough corrective control produced by optimization functions withoutviolation of security constraints. This is referred to as securityconstrained optimization of operation. Such an optimization is describedin the U.S. Pat. No. 5,081,591 dated Jan. 13, 1992: “Optimizing ReactivePower Distribution in an Industrial Power Network”, where the presentinvention can be embodied by replacing the step nos. 56 and 66 each by astep of constant gain matrices [Yf] and [Ye], and replacing steps of“Exercise Newton-Raphson Algorithm” by steps of “Exercise PSDL-YY1 orPSDL-YY2 or YPL-1 or YPL-2 or SZPL or GSPL Computation” in places ofsteps 58 and 68. This is just to indicate the possible embodiment of thepresent invention in optimization functions like in many othersincluding state estimation function. However, invention is being claimedthrough a simplified embodiment without optimization function as in FIG.8 in this application. The inventive steps-42 and -44 in FIG. 8 aredifferent than those corresponding steps-56, and -58, which constitute awell known Newton-Raphson loadflow method, and were not inventive evenin U.S. Pat. No. 5,081,591.

In FIG. 8, function step 12 provides stored impedance values of eachnetwork component in the system. This data is modified in a functionstep 14, which contains stored information about the open or closestatus of each circuit breaker. For each breaker that is open, thefunction step 14 assigns very high impedance to the associated line ortransformer. The resulting data is than employed in a function step 16to establish an admittance matrix for the power network. The dataprovided by function step 12 can be input by the computer operator fromcalculations based on measured values of impedance of each line andtransformer, or on the basis of impedance measurements after the powernetwork has been assembled.

Each of the transformers T1 and T2 in FIG. 9 is a tap changingtransformer having a plurality of tap positions each representing agiven transformation ratio. An indication of initially assignedtransformation ratio for each transformer is provided by function step18 in FIG. 8.

The indications provided by function steps 14, and 22 are supplied to afunction step 42 in which constant gain matrices [Yf] and [Ye], or [Y]or [Z] or [C]⁻¹ of any of the invented PSDL-YY1 or PSDL-YY2 or YPL-1 orYPL-2 or SZPL or GSPL models are constructed, factorized or inverted andstored. The coefficient matrices [Yf] and [Ye], or [C] or [C]⁻¹ or [Z]are conventional tools employed for solving PSDL-1 or PSDL-2 or CPL-1 orCPL-2 or SZPL models defined by equations {(28) and (29)} or {(30) and(31)} or (56) or (58) or (69) or (70) of a power system. [C] is the mostgeneral representation of all possible matrices involved in the solutionof linear and non-linear simultaneous algebraic equations. [C] could bethe Jacobian, approximated Jacobian, constant Jacobian, approximatedconstant decoupled Jacobian in case of Newton-Raphson based approaches.It could be the coefficient matrix, approximated coefficient matrix,constant coefficient matrix, approximated constant decoupled coefficientmatrix in case of Patel Numerical Method (PNM) based approachesdescribed as preferred embodiments in this application. [C]⁻¹ when fullyinverted is the full matrix. However, it can be made sparse by storingand processing only selected elements, and it becomes approximation offully inverted [C]⁻¹.

Indications of initial reactive power, or Q on each node, based oninitial calculations or measurements, are provided by a function step 22and these indications are used in function step 24, to assign a Q levelto each generator and motor. Initially, the Q assigned to each machinecan be the same as the indicated Q value for the node to which thatmachine is connected.

An indication of measured real power, P, on each node is supplied byfunction step 32. Indications of assigned/specified/scheduled/setgenerating plant loads that are constituted by known program areprovided by function step 34, which assigns the real power, P, load foreach generating plant on the basis of the total P, which must begenerated within the power system. The value of P assigned to each powerplant represents an economic optimum, and these values represent fixedconstraints on the variations, which can be made by the system accordingto the present invention. The indications provided by function steps 32and 34 are supplied to function step 36 which adjusts the P distributionon the various plant nodes accordingly. Function step 38 assigns initialapproximate or guess solution to begin iterative method of loadflowcomputation, and reads data file of operating limits on power networkcomponents, such as maximum and minimum reactive power generationcapability limits of PV-nodes generators.

The indications provided by function steps 24, 36, 38 and 42 aresupplied to function step 44 where inventive PSDL-YY1 or PSDL-YY2 orYPL-1 or YPL-2 or SZPL or GSPL model solution is carried out, theresults of which appear in function step 46. The loadflow computationyields voltage magnitudes and voltage angles at PQ-nodes, real andreactive power generation by the reference/slack/swing node generator,voltage angles and reactive power generation indications at PV-nodes,and transformer turns ratio or tap position indications for tap changingtransformers. The system stores in step 44 a representation of thereactive capability characteristic of each PV-node generator and thesecharacteristics act as constraints on the reactive power that can becalculated for each PV-node generator for indication in step 46. Theindications provided in step 46 actuate machine excitation control andtransformer tap position control. All the loadflow computation methodsusing inventive PSDL-YY1 or PSDL-YY2 or YPL-1 or YPL-2 or SZPL or GSPLcomputation models can be used to affect efficient and reliable voltagecontrol in power systems as in the process flow diagram of FIG. 8.

Particularly inventive PSDL-YY1 or PSDL-YY2 or CPL-1 or CPL-2 or SZPL orGSPL models in terms of equations for determining elements of vectors[RI′], [II′], [ΔRI′], [ΔII′], [I], [ΔI] and elements of coefficientmatrices [Yf] and [Ye], or [C] or [Z] are described followed bycomputation steps of corresponding methods are described.

The presence of values of known/given/specified/scheduled/set quantitiesin the diagonal elements of the coefficient matrix [Yf] and [Ye], or [C]or [Z], which takes different form for different methods, is broughtabout by such formulation of loadflow equations. The said quantities inthe diagonal elements in the coefficient matrices improved convergenceand the reliability of obtaining converged loadflow computation.

The slack-start is to use the same voltage magnitude and angle as thoseof the reference/slack/swing node as the initial guess solution estimatefor initiating the iterative loadflow computation. With thespecified/scheduled/set voltage magnitudes, PV-node voltage magnitudesare adjusted to their known values after the first P-θ iteration. Thisslack-start saves almost all effort of mismatch calculation in the firstP-f iteration. It requires only shunt flows from each node to ground tobe calculated at each node, because no flows occurs from one node toanother because they are at the same voltage magnitude and angle.

Patel Numerical Method

The following inventions are based on the Patel Numerical Method-1(PNM-1) originally propounded by this inventor in 2007 in hisinternational patent application no. PCT/CA2007/001537 and consequentgranted patents CA 2661753 and U.S. Pat. No. 8,315,742. The inventedclass of methods of forming/defining and solving loadflow computationmodels of a power network are the methods that organize a set ofnonlinear algebraic equations in linear form as a product of coefficientmatrix and unknown vector on one side and all other terms on the otherside or the corresponding mismatch vector on the other side, and thensolving the linear matrix equation for unknown vector in an iterativefashion.

Propounding Statement of Patel Numerical Method-2 (PNM-2)

-   1. Organize linear or nonlinear equations as mismatch functions    equated to zero.-   2. In each of the mismatch functions, club any term with known    quantities or value into a diagonal term with simple algebraic    manipulations.-   3. Express a vector of the mismatch functions as a product of a    coefficient matrix and a vector of unknown variables, which can    sometimes be treated as a correction vector of unknown variables.-   4. Equate the vector of mismatch functions to the product of the    coefficient matrix and the vector of unknown variables or the    correction vector of unknown variables to be calculated.-   5. Solve such a matrix equation by iterations for the vector of    unknown variables or the correction vector of unknown variables    using evaluation of the vector of mismatch functions with guess    values of unknown variables to begin with, and inverting or    factoring the coefficient matrix.

Preliminary investigations suggest that Patel Numerical Method maypotentially produce monotonous convergence, and therefore may beamenable to acceleration factors unlike Newton-Raphson method.

Patel Loadflow-1 (PL-1)

The PL-1 Model comprises eqns. (1) to (9)

$\begin{matrix}{\mspace{79mu} {\begin{pmatrix}{RI} \\\; \\{II}\end{pmatrix} = {\begin{pmatrix}\; \\C \\\;\end{pmatrix}\begin{pmatrix}f \\\; \\e\end{pmatrix}}}} & (1) \\{\mspace{79mu} {{\begin{pmatrix}f \\\; \\e\end{pmatrix}^{({r + 1})} = {\begin{pmatrix}\; \\C \\\;\end{pmatrix}^{- 1}\begin{pmatrix}{RI} \\\; \\{II}\end{pmatrix}^{(r)}}}\mspace{20mu} {{Where},}}} & (2) \\{{RI}_{p} = {{\left( {{e_{p}{PSH}_{p}} + {f_{p}{QSH}_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)} = {{- \left\lbrack {{\left( {B_{pp} + b_{p}} \right)f_{p}} + {\sum\limits_{q > p}{B_{pq}f_{q}}}} \right\rbrack} + \left\lbrack {{\left( {G_{pp} + g_{p}} \right)e_{p}} + {\sum\limits_{q > p}{G_{pq}e_{q}}}} \right\rbrack}}} & (3) \\{{II}_{p} = {{\left( {{e_{p}{QSH}_{p}} + {f_{p}{PSH}_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)} = {{- \left\lbrack {{\left( {G_{pp} + g_{p}} \right)f_{p}} + {\sum\limits_{q > p}{G_{pq}f_{q}}}} \right\rbrack} - \left\lbrack {{\left( {B_{pp} + b_{p}} \right)e_{p}} + {\sum\limits_{q > p}{B_{pq}e_{q}}}} \right\rbrack}}} & (4) \\{\mspace{79mu} {\begin{pmatrix}\; \\C \\\;\end{pmatrix} = \begin{pmatrix}{Bf} & {Ge} \\\; & \; \\{Gf} & {Be}\end{pmatrix}}} & (5) \\{\mspace{79mu} {{Bf}_{pq} = {{Be}_{pq} = {{{- B_{pq}}\mspace{14mu} {Bf}_{pp}} = {{Be}_{pp} = {- \left( {B_{pp} + b_{p}} \right)}}}}}} & (6) \\{\mspace{79mu} {{Gf}_{pq} = {{Ge}_{pq} = {{{- G_{pq}}\mspace{14mu} {Gf}_{pp}} = {{- {Ge}_{pp}} = {- \left( {G_{pp} + g_{p}} \right)}}}}}} & (7)\end{matrix}$

The equations (1) to (7) represents linearized global solution of thenonlinear loadflow equations. Local nonlinearity can be handled byintroduction of self-iterations as per equations (8) to (9).

[f _(p) ^((sr+1))]^((r+1))=[(RI _(p) /Bf _(pp))^((sr))]^((r))  (8)

[e _(p) ^((sr+1))]^((r+1))=[(II _(p) /Be _(pp))^((sr))]^((r))  (9)

Equations (8) to (9) are solved independently for each node, and can beperformed simultaneously in parallel for all the nodes. Equations (2)and {(8) and (9)} are solved in sequence. In other words linear globalsolution followed by non-linear local (nodal) solution byself-iterations, or non-linear local (nodal) solution by self-iterationsfollowed by linear global solution.

Patel Loadflow-2 (PL-2)

The PL-2 model comprises eqns. {(11) and (12)} or (14), (5), (15) to(20), and {(21) to (24)} or {(25) to (26)}.

$\begin{matrix}{\mspace{79mu} {\begin{pmatrix}{\Delta \; {RI}} \\\; \\{\Delta \; {II}}\end{pmatrix} = {\begin{pmatrix}\; \\C \\\;\end{pmatrix}\begin{pmatrix}{\Delta \; f} \\\; \\{\Delta \; e}\end{pmatrix}}}} & (10) \\{\mspace{79mu} {\begin{pmatrix}{\Delta \; f} \\\; \\{\Delta \; e}\end{pmatrix}^{({r + 1})} = {\begin{pmatrix}\; \\C \\\;\end{pmatrix}^{- 1}\begin{pmatrix}{\Delta \; {RI}} \\\; \\{\Delta \; {II}}\end{pmatrix}^{(r)}}}} & (11) \\{\mspace{79mu} {\begin{pmatrix}f \\\; \\e\end{pmatrix}^{({r + 1})} = {\begin{pmatrix}f \\\; \\e\end{pmatrix}^{(r)} + \begin{pmatrix}{\Delta \; f} \\\; \\{\Delta \; e}\end{pmatrix}^{({r + 1})}}}} & (12) \\{\mspace{79mu} {\begin{pmatrix}{\Delta \; {RI}} \\\; \\{\Delta \; {II}}\end{pmatrix} = {\begin{pmatrix}\; \\C \\\;\end{pmatrix}\begin{pmatrix}f \\\; \\e\end{pmatrix}}}} & (13) \\{\mspace{79mu} {\begin{pmatrix}f \\\; \\{\; e}\end{pmatrix}^{({r + 1})} = {\begin{pmatrix}\; \\C \\\;\end{pmatrix}^{- 1}\begin{pmatrix}{\Delta \; {RI}} \\\; \\{\Delta \; {II}}\end{pmatrix}^{(r)}\mspace{14mu} {W{here}}}}} & (14) \\{{\Delta \; {RI}_{p}} = {{\left( {{e_{p}{PSH}_{p}} + {f_{p}{QSH}_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)} + \left\lbrack {{\left( {B_{pp} + b_{p}} \right)f_{p}} + {\sum\limits_{q > p}{B_{pq}f_{q}}}} \right\rbrack - \mspace{130mu} \left\lbrack {{\left( {G_{pp} + g_{p}} \right)e_{p}} + {\sum\limits_{q > p}{G_{pq}e_{q}}}} \right\rbrack}} & (15) \\{{\Delta \; {RI}_{p}} = {\left\lbrack {{\left\{ {\left( {B_{pp} + b_{p}} \right) + {{QSH}_{p}/\left( {e_{p}^{2} + f_{p}^{2}} \right)}} \right\} f_{p}} + {\sum\limits_{q > p}{B_{pq}f_{q}}}} \right\rbrack - \mspace{95mu} \left\lbrack {{\left\{ {\left( {G_{pp} + g_{p}} \right) - {{PSH}_{p}/\left( {e_{p}^{2} + p_{p}^{2}} \right)}} \right\} e_{p}} + {\sum\limits_{q > p}{G_{pq}e_{q}}}} \right\rbrack}} & (15) \\{\mspace{79mu} {{\Delta \; {RI}_{p}} = {\left( {{e_{p}\Delta \; P_{p}} + {f_{p}\Delta \; Q_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)}}} & (15) \\{{\Delta \; {RI}_{p}} \approx {\left\lbrack {\left( {{e_{p}{PSH}_{p}} + {f_{p}{QSH}_{p}}} \right)/\left( {e_{s}^{2} + f_{s}^{2}} \right)} \right\rbrack - \left\lbrack {\left( {{e_{p}{PSH}_{p}} + {f_{p}{QSH}_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\rbrack}} & (15) \\{{\Delta \; {II}_{p}} = {{\left( {{e_{p}{QSH}_{p}} - {f_{p}{PSH}_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)} + \left\lbrack {{\left( {G_{pp} + g_{p}} \right)f_{p}} + {\sum\limits_{q > p}{G_{pq}f_{q}}}} \right\rbrack + \mspace{140mu} \left\lbrack {{\left( {B_{pp} + b_{p}} \right)e_{p}} + {\sum\limits_{q > p}{B_{pq}e_{q}}}} \right\rbrack}} & (16) \\{{\Delta \; {II}_{p}} = {\left\lbrack {{\left\{ {\left( {G_{pp} + g_{p}} \right) - {{PSH}_{p}/\left( {e_{p}^{2} + f_{p}^{2}} \right)}} \right\} f_{p}} + {\sum\limits_{q > p}{G_{pq}f_{q}}}} \right\rbrack + \mspace{101mu} \left\lbrack {{\left\{ {\left( {B_{pp} + b_{p}} \right) + {{QSH}_{p}/\left( {e_{p}^{2} + f_{p}^{2}} \right)}} \right\} e_{p}} + {\sum\limits_{q > p}{B_{pq}e_{q}}}} \right\rbrack}} & (16) \\{\mspace{79mu} {{\Delta \; {II}_{p}} = {\left( {{e_{p}\Delta \; Q_{p}} - {f_{p}\Delta \; P_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)}}} & (16) \\{{\Delta \; {II}_{p}} \approx {\left\lbrack {\left( {{e_{p}{QSH}_{p}} - {f_{p}{PSH}_{p}}} \right)/\left( {e_{s}^{2} + f_{s}^{2}} \right)} \right\rbrack - \mspace{346mu} \left\lbrack {\left( {{e_{p}{QSH}_{p}} - {f_{p}{PSH}_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\rbrack}} & (16) \\{\mspace{79mu} {{Bf}_{pq} = {{Be}_{pq} = B_{pq}}}} & (17) \\{\mspace{79mu} {{Gf}_{pq} = {{- {Ge}_{pq}} = G_{pq}}}} & (18) \\{{Bf}_{pp} = {{Be}_{pp} = {{\left\lbrack {B_{pp} + b_{p}} \right\rbrack + {{QSH}_{p}/\left( {e_{p}^{2} + f_{p}^{2}} \right)}} \approx {\left\lbrack {B_{pp} + b_{p}} \right\rbrack + {{QSH}_{p}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}}}}} & (19) \\{{Gf}_{pp} = {{- {Ge}_{pp}} = {{\left\lbrack {G_{pp} + g_{p}} \right\rbrack - {{PSH}_{p}/\left( {e_{p}^{2} + f_{p}^{2}} \right)}} \approx {\left\lbrack {G_{pp} + g_{p}} \right\rbrack - {{PSH}_{p}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}}}}} & (20)\end{matrix}$

Equations (15) and (16) provides alternative expressions of real andimaginary current mismatches where ΔQ_(p)=0.0 at PV-nodes. Analternative definition of PL-2 model can be provided by defining ΔRI_(p)of (15) and ΔII_(p) of (16) as the subtraction of the terms containingspecified values from the calculated values that would make ΔRI_(D) andΔII_(D) defined by eqns. (15) and (16) and elements of [C] defined byeqns. (17) to (20) negative.

It can be seen that diagonal elements of the coefficient matrix [C] arechanging with changing values of (e_(p) ²+f_(p) ²), and therefore,requiring time consuming re-factorization of [C] in each iteration. Toavoid re-factorization, it is proposed to make [C] constant by using(e_(s) ²+f_(s) ²), the slack-node voltage values, instead of (e_(p)²+f_(p) ²) in equations (19) and (20) requiring factorization of [C]only once in the beginning of the iteration process.

The equations (10) to (20) represents linearized global solution of thenonlinear loadflow equations. Local nonlinearity can be handled byintroduction of self-iterations as per equations {(21) to (24)} or {(25)to (26)}. It is possible to expand in detail all equations involvingself iterations as in equations (21), (23), (25), (26), (54), (55),(66), (67) etc. in the following.

[Δf _(p) ^((sr+1))]^((r+)1)=[(ΔRI _(p) /Bf _(pp))^((sr))]^((r))  (21)

[f _(p) ^((sr+1))]^((r+1))=[f _(p) ^((sr))]^((r))+[Δf _(p)^((sr+1))]^((r+1))  (22)

[Δ_(ep) ^((sr+1))]^((r+1))=[(ΔII _(p) /Be _(pp))^((sr))]^((r))  (23)

[e _(p) ^((sr+1))]^((r+1))=[e _(p) ^((sr))]^((r))[Δe _(p)^((sr+1))]^((r+1))  (24)

Equations {(21) to (24)} or {(25) to (26)} are solved independently foreach node, and can be performed simultaneously in parallel for all thenodes. Equations {(11) and (12)} or (14), and {(21) to (24)} or {(25)and (26)} are solved in sequence. In other words linear global solutionfollowed by non-linear local (nodal) solution by self-iterations, ornon-linear local (nodal) solution by self-iterations followed by linearglobal solution.

[f _(p) ^((sr+1))]^((r+1))=[(ΔRI _(p) /Bf _(pp))^((sr))]^((r))  (25)

[e _(p) ^((sr+1))]^((r+1))=[(ΔII _(p) /Be _(pp))^((sr))]^((r))  (26)

Patel Super Decoupled Loadflow (PSDL)

In a class of super decoupled loadflow models, each super decoupledloadflow model comprises a system of equations {(28) and (29)} or {(30)and (31)} differing in the definition of elements of [ΔRI′], [ΔII′],[RI′], [II′], and [Yf] and [Ye]. It is a system of equations for theseparate calculation of imaginary part of and real part of complexvoltage or its corrections. [C′] is the transformed coefficient matrix.

$\begin{matrix}{\begin{pmatrix}\; \\C^{\prime} \\\;\end{pmatrix} = \begin{pmatrix}{Yf} & {\; 0} \\\; & \; \\0 & {Ye}\end{pmatrix}} & (27) \\{\left\lbrack {\Delta \; {RI}^{\prime}} \right\rbrack = {\lbrack{Yf}\rbrack \left\lbrack {\Delta \; f} \right\rbrack}} & (28) \\{\left\lbrack {\Delta \; {II}^{\prime}} \right\rbrack = {\lbrack{Ye}\rbrack \left\lbrack {\Delta \; e} \right\rbrack}} & (29) \\{\left\{ {\left\lbrack {\Delta \; {RI}^{\prime}} \right\rbrack \mspace{14mu} {{or}\mspace{14mu}\left\lbrack {RI}^{\prime} \right\rbrack}} \right\} = {\lbrack{Yf}\rbrack \lbrack f\rbrack}} & (30) \\{\left\{ {\left\lbrack {\Delta \; {II}^{\prime}} \right\rbrack \mspace{14mu} {{or}\mspace{14mu}\left\lbrack {II}^{\prime} \right\rbrack}} \right\} = {\lbrack{Ye}\rbrack \lbrack e\rbrack}} & (31)\end{matrix}$

Successive (1f, 1e) Iteration Scheme

In this scheme {(28) and (29)} or {(30) and (31)} are solved alternatelywith intermediate updating. Each iteration involves one calculation of{[ΔRI′] or [RI′]} and {[Δf] or [f]} to update [f] and then onecalculation of {[ΔII′] or [II′]} and {[Δe] or [e]} to update [e]. Thesequence of relations {(32) to (35)} or {(36) to (37)} depicts thescheme.

[Δf]=[Yf]⁻¹[ΔRI′]  (32)

[f]=[f]+[Δf]  (33)

[Δe]=[Ye]⁻¹[ΔII′]  (34)

[e]=[e]+[Δe]  (35)

[f]=[Yf]⁻¹{[ΔRI′] or [RI′]}  (36)

[e]=[Ye]⁻¹{[ΔII′] or [II′]}  (37)

The scheme involves solution of system of equations {(28) and (29)} or{(30) and (31)} in an iterative manner depicted in the sequence ofrelations {(32) to (35)} or {(36) to (37)}. This scheme requirescalculation for each half iteration because {[ΔRI′] and [ΔII′]} or{[RI′] and [II′]} is calculated always using the most recent imaginarypart of and real part of complex voltage values, and it is blockGauss-Seidel approach. The scheme is block successive, which impartsincreased stability to the solution process, and it in turn improvesconvergence and increases the reliability of obtaining solution.

Patel Super Decoupled Loadflow-1 (PSDL-YY1)

The PSDL-YY1 model comprises equations{(32) to (35)} or {(36) to (37)} &(38) to (50).

Where,

$\begin{matrix}\begin{matrix}{{Yf}_{pq} = {{Ye}_{pq} = \left\{ \begin{matrix}{Y_{pq}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} \leq 3.0} \\{\left( {B_{pq} + {0.9\left( {Y_{pq} - B_{pq}} \right)}} \right)\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} > 3.0} \\{B_{pq}\text{:}} & {{{for}\mspace{14mu} {branches}\mspace{14mu} {connected}\mspace{14mu} {between}}\mspace{25mu}} \\\; & {{{two}\mspace{20mu} {PV}\text{-}{nodes}\mspace{14mu} {or}\mspace{14mu} a}\mspace{14mu}} \\\; & {{PV}\text{-}{node}\mspace{14mu} {and}\mspace{14mu} {the}\mspace{14mu} {slack}\text{-}{node}}\end{matrix} \right.}} \\{{Yf}_{pp} = {{Ye}_{pp} = {b_{p}^{\prime} + {\sum\limits_{q > p}{- {Yf}_{pq}}}}}}\end{matrix} & \begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}(38) \\\;\end{matrix} \\\;\end{matrix} \\\;\end{matrix} \\\;\end{matrix} \\\;\end{matrix} \\(39)\end{matrix} \\\;\end{matrix} \\{\begin{matrix}{b_{p}^{\prime} = {{\left( {{{QSH}_{p}{Cos}\; \Phi_{p}} - {{PSH}_{p}{Sin}\; \Phi_{p}}} \right)/\left( {e_{s}^{2} + f_{s}^{2}} \right)} + {b_{p}{Cos}\; \Phi_{p}\text{:}}}} & {{at}\mspace{14mu} {PQ}\text{-}{node}} & \mspace{34mu} \\{b_{p}^{\prime} = {{Q_{p\; 0}/\left( {e_{s}^{2} + f_{s}^{2}} \right)} + {b_{p}\text{:}}}} & {{at}\mspace{14mu} {PV}\text{-}{node}} & \;\end{matrix}\left( {Q_{p\; 0} - {{calculated}\mspace{14mu} {at}\mspace{14mu} {initial}\mspace{14mu} {estimate}\mspace{14mu} {solution}}} \right)} & \begin{matrix}(40) \\(41)\end{matrix} \\\begin{matrix}{{\Delta \; {{RI}_{p}}^{\prime}} = {{\Delta \; {RI}_{p}{Cos}\; \Phi_{p}} + {\Delta \; {II}_{p\;}{Sin}\; \Phi_{p}\text{:}}}} & {{for}\mspace{14mu} {PQ}\text{-}{nodes}} \\{{\Delta \; {{RI}_{p}}^{\prime}} = {{\left( {{e_{p}{\Delta P}_{p}^{\prime}} + {f_{p}\Delta \; Q_{p}^{\prime}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)}\text{:}}} & {{for}\mspace{14mu} {PQ}\text{-}{nodes}} \\{{\Delta \; {II}_{p}^{\prime}} = {{\Delta \; {II}_{p}{Cos}\; \Phi_{p}} - {\Delta \; {RI}_{p}{Sin}\; \Phi_{p}\text{:}}}} & {{for}\mspace{14mu} {PQ}\text{-}{nodes}} \\{{\Delta \; {II}_{p}^{\prime}} = {{\left( {{e_{p}\Delta \; Q_{p}^{\prime}} - {f_{p}\Delta \; P_{p}^{\prime}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)}\text{:}}} & {{for}\mspace{14mu} {PQ}\text{-}{nodes}} \\{{\Delta \; P_{p}^{\prime}} = {{\Delta \; P_{p}{Cos}\; \Phi_{p}} + {\Delta \; Q_{p}{Sin}\; \Phi_{p}\text{:}}}} & {{for}\mspace{14mu} {PQ}\text{-}{nodes}} \\{{\Delta \; Q_{p}^{\prime}} = {{\Delta \; Q_{p}{Cos}\; \Phi_{p}} - {\Delta \; P_{p}{Sin}\; \Phi_{p}\text{:}}}} & {{for}\mspace{14mu} {PQ}\text{-}{nodes}} \\{{\Delta \; {RI}_{p}} = {{\left( {e_{p}\Delta \; P_{p}} \right)/\left\lbrack {K_{p}\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\rbrack}\text{:}}} & {{for}\mspace{14mu} {PV}\text{-}{nodes}} \\{{\Delta \; {II}_{p}} = {{\left( {{- f_{p}}\Delta \; P_{p}} \right)/\left\lbrack {K_{p}\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\rbrack}\text{:}}} & {{for}\mspace{14mu} {PV}\text{-}{nodes}} \\{{{Cos}\; \Phi_{p}} = {{\left\lbrack {B_{pp}/\left. \sqrt{}\left( {G_{pp}^{2} + B_{pp}^{2}} \right) \right.} \right\rbrack } \geq {{Cos}\left( {{0{^\circ}\mspace{14mu} {to}}\mspace{14mu} - {90{^\circ}}} \right)\text{:}}}} & {{to}\mspace{14mu} {be}\mspace{14mu} {determined}\mspace{14mu} {experimentally}} \\{{{Sin}\; \Phi_{p}} = {{- {\left\lbrack {G_{pp}/\left. \sqrt{}\left( {G_{pp}^{2} + B_{pp}^{2}} \right) \right.} \right\rbrack }} \geq {{Sin}\left( {{0{^\circ}\mspace{14mu} {to}}\mspace{14mu} - {90{^\circ}}} \right)\text{:}}}} & {{to}\mspace{14mu} {be}\mspace{14mu} {determined}\mspace{14mu} {experimentally}} \\{K_{p} = {\left( {B_{pp}/{\sum\limits_{q > p}{- {Yf}_{pp}}}} \right)}} & \;\end{matrix} & \begin{matrix}(42) \\\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}(42) \\(43)\end{matrix} \\(43)\end{matrix} \\(44)\end{matrix} \\(45)\end{matrix} \\(46)\end{matrix} \\\begin{matrix}\begin{matrix}\begin{matrix}(47) \\(48)\end{matrix} \\(49)\end{matrix} \\(50)\end{matrix}\end{matrix}\end{matrix}\end{matrix}$

Super Super Decoupled Loadflow (SSDL-YY)

Two new versions of SSDL-YY are provided. One is Hybrid SSDL-YY(HSSDL-YY) and the other is Efficient SSDL-YY (ESSDL-YY). The HSSDLmodel comprises eqns. (32) to (35), (38a), (38b), (39a), (39b), (40a),(40b), (41a), (41b), (42) to (45), (46a), (47a), and (48) to (50). TheESSDL-YY model comprises eqns. (32) to (35), (38c), (39a), (39b), (40a),(40b), {(41c) and (41d)} where QSH_(p) replaced by Q_(p0) calculatedvalue at initial estimate at PV-nodes, {(42) and (43)} with approximateversions of {(15) and (16)} where QSH_(p) replaced by Q_(p) (calculatedvalue) at PV-nodes, and {(48) and (49)}.

$\begin{matrix}{{Yf}_{pq} = \left( \begin{matrix}{{- Y_{pq}}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} \leq 3.0} \\{{- \left( {B_{pq} + {0.9\left( {Y_{pq} - B_{pq}} \right)}} \right)}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} > 3.0} \\{{- B_{pq}}\text{:}} & \begin{matrix}{{for}\mspace{14mu} {branches}\mspace{14mu} {connected}\mspace{14mu} {between}\mspace{14mu} {two}} \\{{{PV}\text{-}{nodes}\mspace{14mu} {or}\mspace{14mu} a\mspace{14mu} {PV}\text{-}{node}\mspace{14mu} {and}}\mspace{11mu}} \\{{the}\mspace{14mu} {slack}\text{-}{node}}\end{matrix}\end{matrix} \right.} & \left( {38a} \right) \\{{Ye}_{pq} = \left( \begin{matrix}{- {Y_{pq}:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} \leq 3.0} \\{{- \left( {B_{pq} + {0.9\left( {Y_{pq} - B_{pq}} \right)}} \right)}:} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} > 3.0}\end{matrix} \right.} & \left( {38b} \right) \\\left( \begin{matrix}{{- Y_{pq}}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} \leq 3.0} \\{{- \left( {B_{pq} + {0.9\left( {Y_{pq} - B_{pq}} \right)}} \right)}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} > 3.0}\end{matrix} \right. & \left( {38c} \right) \\{{Yf}_{pp} = {{bf}_{p}^{\prime} + {\sum\limits_{q > p}{- {Yf}_{pq}}}}} & \left( {39a} \right) \\{\mspace{11mu} {{Ye}_{pp} = {{be}_{p}^{\prime} + {\sum\limits_{q > p}{- {Ye}_{pq}}}}}} & \left( {39b} \right) \\\begin{matrix}{{bf}_{p}^{\prime} = {{{+ \left( {{{QSH}_{p\;}{Cos}\; \Phi_{p}} - {{PSH}_{p}{Sin}\mspace{11mu} \Phi_{p}}} \right)}/\left( {e_{s}^{2} + f_{s}^{2}} \right)} - {b_{p}{Cos}\; \Phi_{p}\text{:}}}} & {{at}\mspace{14mu} {PQ}\text{-}{node}} \\{{be}_{p}^{\prime} = {{{- \left( {{{QSH}_{p\;}{Cos}\; \Phi_{p}} - {{PSH}_{p}{Sin}\mspace{11mu} \Phi_{p}}} \right)}/\left( {e_{s}^{2} + f_{s}^{2}} \right)} - {b_{p}{Cos}\; \Phi_{p}\text{:}}}} & {{at}\mspace{14mu} {PQ}\text{-}{node}} \\{{bf}_{p}^{\prime} = {0.0\text{:}}} & {{at}\mspace{14mu} {PV}\text{-}{node}} \\{{be}_{p}^{\prime} = {10.0^{10}\left( {{say},{{it}\mspace{14mu} {is}\mspace{14mu} {chosen}\mspace{14mu} {very}\mspace{14mu} {large}\mspace{14mu} {value}}} \right)\text{:}}} & {{at}\mspace{14mu} {PV}\text{-}{node}} \\{{bf}_{p}^{\prime} = {{{+ \left( {{{QSH}_{p}{Cos}\; \Phi_{p}} - {{PSH}_{p}{Sin}\; \Phi_{p}}} \right)}/\left( {e_{s}^{2} + f_{s}^{2}} \right)} - {b_{p}{Cos}\; \Phi_{p}\text{:}}}} & {{at}\mspace{14mu} {PV}\text{-}{node}} \\{{be}_{p}^{\prime} = {{{- \left( {{{QSH}_{p}{Cos}\; \Phi_{p}} - {{PSH}_{p}{Sin}\; \Phi_{p}}} \right)}/\left( {e_{s}^{2} + f_{s}^{2}} \right)} - {b_{p}{Cos}\; \Phi_{p}\text{:}}}} & {{at}\mspace{14mu} {PV}\text{-}{node}} \\{{\Delta \; {RI}_{p}} = {\Delta \; {P_{p}/\left( {K_{p}V_{p}^{2}} \right)}\text{:}}} & {{at}\mspace{14mu} {PV}\text{-}{node}} \\{{\Delta \; {II}_{p}} = {0.0\text{:}}} & {{at}\mspace{14mu} {PV}\text{-}{node}}\end{matrix} & \begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\begin{matrix}\left( {40a} \right) \\\left( {40b} \right)\end{matrix} \\\left( {41a} \right)\end{matrix} \\\left( {41b} \right)\end{matrix} \\\left( {41c} \right)\end{matrix} \\\left( {41d} \right)\end{matrix} \\\left( {46a} \right)\end{matrix} \\\left( {47a} \right)\end{matrix}\end{matrix}$

Branch admittance magnitude in (38), (38a), (38b), (38c) is of the samealgebraic sign as its susceptance. Rotation angles are to be determinedas per (48) and (49), and could be restricted to the maximum anywhere −0to −90 degrees to be determined experimentally. There can be manypossible variations of PSDL, HSSDL, and ESSDL models, and the onevariation being their generalized versions PSDL-B′B′, HSSDL-B′B′, andESSDL-B′B′ where B′ symbolizes suceptance matrix transformed,B′_(pq)=B_(pq)+G_(pq) tan Φ_(pq) and tan Φ_(pq)=G_(pq)/B_(pq). Also, thetwo versions PSDL-YY and PSDL-B′B′, HSSDL-YY and HSSDL-B′B′, andESSDL-YY and ESSDL-B′B′ can be mixed in any possible combination.Corresponding transformed diagonal elements B_(pp)′ and transformedmismatches can easily be determined.

Slack-Start

Slack-Start is use of the same voltage magnitude and angle as those ofthe slack-node for all nodes as an initial guess solution. With thespecified magnitudes, PV-nodes voltage magnitudes are adjusted to theirknown values after the first half iteration. This start procedurereferred to as the slack-start, saves almost all effort of mismatchcalculation in the first P-f iteration as it requires only shunt flowsto be calculated at each node.

where, K_(p) is defined in equation (50) which is initially restrictedto the minimum value of 0.75 determined experimentally; however itsrestriction is lowered to the minimum value of 0.6 when its average overall less than 1.0 values at PV nodes is less than 0.6.

In super decoupled loadflow models [Yf] and [Ye] are real, sparse,symmetrical and built only from network elements. Since they areconstant, they need to be factorized once only at the start of thesolution. Equations {(28) and (29)} or {(30) and (31)} are to be solvedrepeatedly by forward and backward substitutions. [Yf] and [Ye] are ofthe same dimensions (m+k)×(m+k) when only a row/column of thereference/slack-node is excluded and both are triangularized using thesame ordering regardless of the node-types.

Unlike the HSSDL and the prior art SSDL (Super Super Decoupled Loadflow,presented at Toronto International Conference—Science and Technology forHumanity—2009, pages: 652-659) methods, the PSDL methods are singlematrix loadflow computations substantially reducing memory requirements,and since all nodes are active in the iterative process implementationsof PQ-node to PV-node and PV-node to PQ-node switching is simple. Thebest possible convergence from non-linearity consideration could beachieved by restricting rotation angle to maximum of −0 to −90 degrees(say, −48 degrees) to be determined experimentally.

The steps of loadflow computation method, PSDL-YY1 method are shown inthe flowchart of FIG. 1. Computation steps of HSSDL method are similar,therefore, they are not given explicitly. Referring to the flowchart ofFIG. 1, different steps are elaborated in steps marked with similarletters in the following. Double lettered steps are the characteristicsteps of PSDL-YY1 method. The words “Read system data” in Step-acorrespond to step-10 and step-20 in FIG. 7, and step-16, step-18,step-24, step-36, step-38 in FIG. 8. All other steps in the followingcorrespond to step-30 in FIG. 7, and step-42, step-44, and step-46 inFIG. 8.

-   a. Read system data and assign an initial approximate solution. If    better solution estimate is not available, set voltage magnitude and    angle of all nodes equal to those of the slack-node, referred to as    the slack-start.-   b. Form nodal admittance matrix, and Initialize iteration count    ITRF=ITRE=r=0-   c. Compute Cosine and Sine of nodal rotation angles using equations    (48), (49), and store them. If Cos Φ_(p)<Cos (0 to −90 degrees, to    be determined experimentally), set Cos Φ_(p)=Cos (say, 0 to −90    degrees to be determined experimentally) and Sin Φ_(p)=Sin (say, 0    to −90 degrees to be determined experimentally).-   dd. Form, factorize, and store (m+k)×(m+k) matrix [Yf] or [Ye] of    {(28) and (29)} or {(30) and (31)} in a compact storage exploiting    sparsity, using equations (38) to (41).-   e. Compute residues [ΔP] at PQ- and PV-nodes and [ΔQ] at only    PQ-nodes. If all are less than the tolerance (c), proceed to step-n.    Otherwise follow the next step.-   ff. Compute the vector of transformed residues [ΔRI′] as in (42) for    PQ-nodes, and using (46) and (50) for PV-nodes.-   gg. Solve {(32) for [Δf]} or {(36) for [f]} and update using,    [f]=[f]+[Δf].-   h. Set voltage magnitudes of PV-nodes equal to the specified values,    and Increment the iteration count ITRF=ITRF+1 and r=(ITRF+ITRE)/2.-   i. Compute residues [ΔP] at PQ- and PV-nodes and [ΔQ] at PQ-nodes    only. If all are less than the tolerance (c), proceed to step-n.    Otherwise follow the next step.-   jj. Compute the vector of transformed residues [ΔII′] as in (43) for    PQ-nodes, and using (47) and (50) for PV-nodes.-   kk. Solve {(34) for [Δe]} or {(37) for [e]} and update using    [e]=[e]+[Δe].-   l. Calculate reactive power generation at PV-nodes and tap positions    of tap-changing transformers. If the maximum and minimum reactive    power generation capability and transformer tap position limits are    violated, implement the violated physical limits and adjust the    loadflow solution by the method like one described in “LTC    Transformers and MVAR violations in the Fast Decoupled Load Flow,    IEEE Trans., PAS-101, No. 9, PP. 3328-3332, September 1982”.-   m. Increment the iteration count ITRE=ITRE+1 and r=(ITRF+ITRE)/2, &    Proceed to step-e.-   n. From calculated values of real and imaginary components of nodal    voltages, reactive power generation at PV-nodes, and tap position of    tap changing transformers, calculate power flows through power    network components.

Patel Super Decoupled Loadflow-2 (PSDL-YY2)

The Patel Super Decoupled Loadflow-2 (PSDL-YY2) model comprisesequations {(32) to (35)} or {(36) to (37)}, {(3), (4), (51), (52),(40c), and (53b)}, or {(42), (43) with approximate versions of (15) and(16), (40), and (53a)}, (39), and {(54) and (55)}. In (3), (4), (15),and (16): QSH_(p) is replaced by Q_(p) (calculated) for PV-nodes, and in(40) QSH_(p) is replaced by Q_(p0) (calculated at initial estimate) forPV-nodes.

Where,

$\begin{matrix}{\mspace{79mu} {{RI}_{p}^{\prime} = {{{RI}_{p}{Cos}\; \Phi_{p}} + {{II}_{p}{Sin}\; \Phi_{p}}}}} & (51) \\{\mspace{79mu} {{II}_{p}^{\prime} = {{{II}_{p}{Cos}\; \Phi_{p}} + {{RI}_{p}{Sin}\; \Phi_{p}}}}} & (52) \\{{Yf}_{pq} = {{Yf}_{pq} = \begin{pmatrix}{Y_{pq}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} \leq 3.0} \\{\left( {B_{pq} + {0.9\left( {Y_{pq} - B_{pq}} \right)}} \right)\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} > 3.0}\end{pmatrix}}} & \left( {53a} \right) \\{{Yf}_{pq} = {{Yf}_{pq} = \begin{pmatrix}{{- Y_{pq}}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} \leq 3.0} \\{{- \left( {B_{pq} + {0.9\left( {Y_{pq} - B_{pq}} \right)}} \right)}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} {r/x}\mspace{14mu} {ratio}} > 3.0}\end{pmatrix}}} & \left( {53b} \right) \\{\mspace{79mu} {b_{p}^{\prime} = {{- b_{p}}{Cos}\; \Phi_{p}\text{:}\mspace{14mu} {at}\mspace{14mu} {PV}\text{-}{nodes}}}} & \left( {40c} \right)\end{matrix}$

All equations other than (54) and (55) of the model PSDL-YY2 representslinearized global solution of the nonlinear loadflow equations. Localnonlinearity can be handled by introduction of self-iterations as perequations (54) to (55).

[f _(p) ^((sr+1))]^((r+1))[{(RI′ _(p) or ΔRI′ _(p))/Yf_(pp)}^((sr))]^((r))  (54)

[e _(p) ^((sr+1))]^((r+1))[{(II′ _(p) or ΔII′ _(p))/Ye_(pp)}^((sr))]^((r))  (55)

Equations (54) to (55) are solved independently for each node, and canbe performed simultaneously in parallel for all the nodes. SuperDecoupled equations {(32) or (36), and (54)} and {(34) or (37), and(55)} are solved in sequence. In other words linear global solutionfollowed by non-linear local (nodal) solution by self-iterations.

The steps of loadflow computation method, PSDL-YY2 method are shown inthe flowchart of FIG. 2. Computation steps of ESSDL method are similar,therefore, they are not given explicitly. Referring to the flowchart ofFIG. 2, different steps are elaborated in steps marked with similarletters in the following. Triple lettered steps are the characteristicsteps of PSDL-YY2 method. The words “Read system data” in Step-acorrespond to step-10 and step-20 in FIG. 7, and step-16, step-18,step-24, step-36, step-38 in FIG. 8. All other steps in the followingcorrespond to step-30 in FIG. 7, and step-42, step-44, and step-46 inFIG. 8.

-   a. Read system data and assign an initial approximate solution    vectors [f0], [e0], and store it. If better solution estimate is not    available, set voltage magnitude to 1.0 pu at load nodes and    specified values at PV-nodes, and angle of all nodes equal to that    of the slack-node, referred to as the flat-start.-   b. Form nodal admittance matrix, and Initialize iteration count    ITRF=ITRE=r=0 and MXDF=MXDE=0.0-   c. Compute Cosine and Sine of nodal rotation angles using equations    (48), (49), and store them. If Cos Φ_(p)<Cos (0 to −90 degrees, to    be determined experimentally), set Cos Φ_(p)=Cos (say, 0 to −90    degrees to be determined experimentally) and Sin Φ_(p)=Sin (say, 0    to −90 degrees to be determined experimentally).-   ddd. Form, factorize, and store (m+k)×(m+k) size matrix [Yf] or [Ye]    of {(28) and (29)} or {(30) and (31)} in a compact storage    exploiting sparsity, using equations {(53a), (39), (40) and (41)} or    {(53b), (39), and (40c)}.-   eee. Compute the vector of transformed residues [ΔRI′] using (42)    {with approximated values of [ΔRI] and [ΔII] from (15) and (16)} or    [RI′] using (51). Use calculated values of Q_(p) in place of QSH_(p)    for PV-nodes, and implement Q-limit violations at PV-node    generators.-   fff. Solve {(32) for [Δf]} or {(36) for [f]}, perform    Self-Iterations for each node using (54), and update using,    [f]=[f]+[Δf].-   g. Take a difference of vectors {[f]−[f0]}, find the maximum of the    element difference and store it in variable ‘MXDF’, and perform    [f0]=[f] and Increment the iteration count ITRF=ITRF+1 and    r=(ITRF+ITRE)/2.-   h1. Are MXDF and MXDE both less than specified tolerance? If it is,    Proceed to step-n or else follow the next step.-   iii. Compute the vector of modified residues [ΔII′] using (43) {with    approximated values [ΔRI] and [ΔII] from (15) and (16)} or [II′]    using (52). Use calculated values of Q_(p) in place of QSH_(p) for    PV-nodes, and implement Q-limit violations at PV-node generators.-   jjj. Solve {(34) for [Δe]} for {(37) for [e]}, perform    Self-Iterations for each node using (55), and update using    [e]=[e]+[Δe].-   k. Take a difference of vectors {[e]−[e0] }, find the maximum of the    element difference and store it in variable ‘MXDE’, and perform    [e0]=[e] and Increment the iteration count ITRE=ITRE+1 and    r=(ITRF+ITRE)/2.-   l1. Are MXDF and MXDE both less than specified tolerance? If it is    not, Proceed to step-eee or else follow the next step.-   n. From calculated values of real and imaginary components of nodal    voltages, reactive power generation at PV-nodes, and tap position of    tap changing transformers, calculate power flows through power    network components.

Coefficient Matrix [C] Based Patel Loadflow (CPL)

Patel Loadflow model can be organized in coefficient matrix [C] basedcomplex form, because it is not involved with any partialdifferentiation of original or mismatch functions. The model constituteseqns. {(57) or (59)}, {(60) to (62)} or {(63) to (65)} or {(60a), (64),and (65a)}, {(66) or (67)}, and (68). It involves one solution of {(57)or (59)} followed by one solution of {(66) or (67)}, or one solution of{(66) or (67)} followed by one solution of {(57) or (59)}. However,{(66) or (67)} constitutes one equation for each node except theSlack-node, and equations for all the nodes can be solved in parallel,just like Gauss numerical method.

[ΔI]=[C][ΔV]  (56)

[ΔV]=[C]⁻¹[ΔI]  (57)

OR

{[ΔI] or [I]}=[C][V]  (58)

[V]=[C]⁻¹{[ΔI] or [I]}  (59)

Where, components of vectors [I] and [ΔI], and matrix [C] are defined inthe following:

$\begin{matrix}{I_{p} = {{\left( {{PSH}_{p} - {jQSH}_{p}} \right)/\left( {e_{p} - {jf}_{p}} \right)} = {\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) = \left\lbrack {{\left( {Y_{pp} + y_{p}} \right)V_{p}} + {\sum\limits_{q > p}{Y_{pq}V_{q}}}} \right\rbrack}}} & (60) \\{{\Delta \; I_{p}} = {{\left( {{SSH}_{p}^{*} - S_{p}^{*}} \right)/V_{p}^{*}} = {{\left\lbrack {\left( {{PSH}_{p} - {jQSH}_{p}} \right) - \left( {P_{p} - {jQ}_{p}} \right)} \right\rbrack/V_{p}^{*}} = {\left( {{\Delta \; P_{p}} - {j\; \Delta \; Q_{p}}} \right)/V_{p}^{*}}}}} & (60) \\{\mspace{79mu} {{\Delta \; I_{p}} = {{\left\lbrack {\left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack V_{p}} - {\sum\limits_{q > p}{Y_{pq}V_{q}}}}}} & (60) \\{{{\Delta \; I_{p}} \approx {\left\lbrack {\left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}} \right\rbrack - \left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\}} \right\rbrack V_{p}}} = {{L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} - {{SSH}_{p}^{*}/V_{p}^{*}}}} & (60) \\{\mspace{79mu} {{{\Delta \; I_{p}} \approx {\left\lbrack {L_{p} - \left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\}} \right\rbrack V_{p}}} = {{L_{p}V_{p}} - {{SSH}_{p}^{*}/V_{p}^{*}}}}} & (60) \\{\mspace{79mu} {C_{pq} = {- Y_{pq}}}} & (61) \\{C_{pp} = {\left\lbrack {\left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)}} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack \approx \left\lbrack {\left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack}} & (62) \\{\mspace{79mu} {C_{pp} = {\left\lbrack {L_{p} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack \mspace{14mu} {OR}}}} & (62) \\{{\Delta \; I_{p}} = {{\left( {S_{p}^{*} - {SSH}_{p}^{*}} \right)/V_{p}^{*}} = {{\left\lbrack {\left( {P_{p} - {jQ}_{p}} \right) - \left( {{PSH}_{p} - {jQSH}_{p}} \right)} \right\rbrack/V_{p}^{*}} = {\left\lbrack {\left( {{- \Delta}\; P_{p}} \right) - {j\left( {{- \Delta}\; Q_{p}} \right)}} \right\rbrack/V_{p}^{*}}}}} & (63) \\{\mspace{79mu} {{\Delta \; I_{p}} = {{\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{*}} \right)} \right\}} \right\rbrack V_{p}} + {\sum\limits_{q > p}{Y_{pq}V_{q}}}}}} & (63) \\{{{\Delta \; I_{p}} \approx {\left\lbrack {\left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\} - \left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}} \right\}} \right\rbrack V_{p}}} = {{{SSH}_{p}^{*}/V_{p}^{*}} - {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}}}} & (63) \\{{{\Delta \; I_{p}} \approx {\left\lbrack {\left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\} - L_{p}} \right\rbrack V_{p}}} = {{{SSH}_{p}^{*}/V_{p}^{*}} - {L_{p}V_{p}}}} & (63) \\{\mspace{79mu} {C_{pq} = Y_{pq}}} & (64) \\{C_{pp} = {\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\}} \right\rbrack \approx \left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}} \right\}} \right\rbrack}} & (65) \\{\mspace{79mu} {C_{pp} = \left\lbrack {\left( {Y_{pp} + y_{p}} \right) - L_{p}} \right\rbrack}} & (65) \\{\mspace{79mu} {C_{pp} = \left( {Y_{pp} + y_{p}} \right)}} & \left( {65a} \right) \\{\mspace{79mu} {\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = \left\lbrack \left( {\Delta \; {I_{p}/C_{pp}}} \right)^{({sr})} \right\rbrack^{(r)}}} & (66) \\{\mspace{79mu} {\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = \left\lbrack \left( {\left( {\Delta \; I_{p}\mspace{14mu} {or}\mspace{14mu} I_{p}} \right)/C_{pp}} \right)^{({sr})} \right\rbrack^{(r)}}} & (67) \\{\mspace{79mu} {{L_{p} = {- \infty}},\ldots \mspace{14mu},{- 1},0,{+ 1},\ldots \mspace{14mu},{+ {\infty \left( {{including}\mspace{14mu} {fractions}} \right)}}}} & (68)\end{matrix}$

The equations (62), (65), and (68) provide elegant formulation fordiagonal elements of the coefficient matrix [C] that suggest a mechanismfor their numerical manipulations particularly useful when diagonaldominance issue arise in the presence of a capacitive series branch oran excessive capacitive compensation at a node. The factor L_(p) ofdifferent value can be applied separately to real and imaginarycomponents of a diagonal element of [C]. Similar developments can beprovided for Patel Super Decoupled Loadflow models and other loadflowmodels. Equations (66) and (67) and their expanded versions can also bewritten with factor L.

It can be seen that diagonal elements of the coefficient matrix [C] arechanging with changing values of V_(p), and therefore, values of (e_(p)²+f_(p) ²) during iteration process requiring time consumingre-factorization of [C] in each iteration. To avoid re-factorization, itis proposed to make [C] constant by using (e_(s) ²+f_(s) ²), theslack-node voltage values, instead of (e_(p) ²+f_(p) ²) in equations(62) and (65) requiring factorization of [C] only once in the beginningof the iteration process.

Coefficient Matrix [C] Based Patel Loadflow-1 (CPL-1)

The steps of loadflow calculation by CPL-1 method are shown in theflowchart of FIG. 3. Referring to the flowchart of FIG. 3, differentsteps are elaborated in steps marked with similar numbers in thefollowing. Double numbered steps are the inventive steps. The words“Read system data” in Step-a correspond to step-10 and step-20 in FIG.7, and step-16, step-18, step-24, step-36, step-38 in FIG. 8. All othersteps in the following correspond to step-30 in FIG. 7, and step-42,step-44, and step-46 in FIG. 8.

-   1. Read system data and assign an initial approximate solution. If    better solution estimate is not available, set voltage magnitude and    angle of all nodes equal to those of the slack-node, referred to as    the slack-start.-   2. Form nodal admittance matrix [Y], and Initialize iteration count    ITR=0-   33. Form, factorize, and store (m+k)×(m+k) size complex coefficient    matrix [C] of {(56) or (58)} in a compact storage exploiting    sparsity, using equations {(61) and (62)} or {(64) and (65)}.-   4. Compute residues [ΔP] at PQ- and PV-nodes and [ΔQ] at only    PQ-nodes. If all are less than the tolerance (c), proceed to step-9.    Otherwise follow the next step.-   55. Compute the vector [ΔI] using equations {(60) or (63)}.-   66. Solve (57) for [ΔV] or (59) for [V], and perform Self-Iterations    for each node using (66) or (67), and update voltage using,    [V]=[V]+[ΔV].-   7. Set voltage magnitudes of PV-nodes equal to the specified values;    Adjust loadflow solution, if generator reactive power and    transformer tap limits are violated; and Increment the iteration    count ITR=ITR+1. Go to step-4-   9. From calculated values of complex nodal voltages and reactive    power generation at PV-nodes, and tap position of tap changing    transformers, calculate power flows through power network    components.

Coefficient Matrix [C] Based Patel Loadflow-2 (CPL-2)

The steps of loadflow calculation by CPL-2 method are shown in theflowchart of FIG. 4. Referring to the flowchart of FIG. 4, differentsteps are elaborated in steps marked with similar numbers in thefollowing. Triple numbered steps are the inventive steps. The words“Read system data” in Step-a correspond to step-10 and step-20 in FIG.7, and step-16, step-18, step-24, step-36, step-38 in FIG. 8. All othersteps in the following correspond to step-30 in FIG. 7, and step-42,step-44, and step-46 in FIG. 8.

-   1. Read system data and assign an initial approximate solution    vector [V0], and store it. If better solution estimate is not    available, set voltage magnitude to 1.0 pu at load nodes and    specified values at PV-nodes, and angle of all nodes equal to that    of the slack-node, referred to as the flat-start.-   2. Form nodal admittance matrix [Y], and Initialize iteration count    ITR=0-   333. Form, factorize, and store (m+k)×(m+k) size complex coefficient    matrix [C] of {(56) or (58)} in a compact storage exploiting    sparsity, using equations {(61) and (62)}, or {(64) and (65)} or    {(64) and (65a)}.-   555. Compute the vector {[I] using (60a)} or {[ΔI]} using    approximated versions of (60) or (63)}. However, use calculated    value Q_(p) instead of QSH_(p) for PV-nodes, and implement violated    Q_(max) or Q_(min) limits of PV-node generators.-   66. Solve {(57) for [ΔV]} or {(59) for [V]}, and perform    Self-Iterations for each node using (66) or (67), and update voltage    using, [V]=[V]+[ΔV].-   777. Take a difference of vectors {[V]−[V0]}, find the maximum of    the element differences and store it in variable ‘MaxΔ’, and perform    [V0]=[V] and ITR=ITR+1.-   8. Is MaxΔ less than specified tolerance? If it is not, Proceed to    step-555 or else follow the next step.-   9. From calculated values of complex nodal voltages, reactive power    generation at PV-nodes, and tap position of tap changing    transformers, calculate power flows through power network    components.    Patel Sparse Inverse Solver (PSIS) and Sparse [C]⁻¹ (Pronounced as    ‘C inverse’, which is sparse [Z]) based Patel Loadflow (SCIPL or    SZPL):

Matrix complex [C] or real [C] is the coefficient matrix based onoriginal equations (functions) or organized as mismatch equations(functions) in the solution of linear or non-linear simultaneousalgebraic equations. Inverses complex [C]⁻¹ and real [C]⁻¹ arecorrespondingly referred to as complex [Z]-matrix and real [Z]-matrix inthis application. The complex [C]⁻¹ and real [C]⁻¹ can also representadmittance matrices respectively complex [Y]⁻¹ and real [Y]⁻¹. Complex[C]⁻¹ and real [C]⁻¹ are generalized representations, wherein real [C]⁻¹can be Newton-Raphson approach based Jacobian [J]⁻¹ and its simplifiedapproximations including inverses of decoupled or Super Decoupledmatrices.

The complex [C] and real [C] are generally sparse matrices wherein manyof its off diagonal elements are zeros. In order to save computationtime and computer storage, processing of off-diagonal elements that arezeros can be avoided by sparsity preserving programming techniques.However, fully inverted complex [Z]-matrix and real [Z]-matrix are fullmatrices wherein no off-diagonal elements are zeros. Therefore, it isproposed to make complex [Z]-matrix and real [Z]-matrix sparse byselectively choosing off-diagonal elements that need to be stored andprocessed, and thereby introducing approximations. There are twoextremes, one is to store and process only one element in each rawcorresponding to the diagonal element introducing maximum approximationand the other is to store and process the diagonal and all theoff-diagonal elements in each row introducing zero approximation. Andthere are many situations in between the two extremes stated in theabove to be determined experimentally depending on the nature of theproblem for optimal use of computational resources (computer time andcomputer storage). In electrical circuits (networks), one situation isto store off-diagonal elements only corresponding to directly connectednodes (level-1 connectivity) to a given node, which is the same sparsityof the matrix complex [C] or real [C]. For a given node, othersituations are to store off-diagonal elements corresponding to directlyconnected nodes (level-1), and directly connected nodes to level-1 nodes(level-2 nodes), and directly connected nodes to level-2 nodes (level-3nodes), and so on. For a given node, the level of outward connectivityis to be determined experimentally to determine number of off-diagonalelement required to be stored and processed in the complex [Z]-matrixand the real [Z]-matrix for efficient and reliable computation.In equations (69) and (70): vectors [V] and [I] are of complex voltageand complex current element components respectively. Vectors [ΔV] and[ΔI] are composed of complex voltage correction and complex currentmismatch components respectively. Voltage and current quantities appearin electrical circuits. Equation (69) corresponds to equation (59),Equation (70) corresponds to equation (58), and relevant quantities aredefined in equations from (60) to (65), wherein complex [Z] becomescomplex [C]⁻¹.Equation (69) corresponds to two super decoupled equations (36) and(37), and equation (70) corresponds to two super decoupled equations(32) and (34). Relevant quantities defined in equations (38) to (50),and (51) to (53). It should be noted that all quantities involved arereal and not complex, wherein real [Z] becomes equivalent to two superdecoupled real matrices [Yf]⁻¹ and [Ye]⁻¹.Application of Newton-Raphson approach to solution of simultaneousnon-linear algebraic equations involves calculation of correction vectorin each iteration and requires updating as in equations (33) and (35) incase of decoupled models.All the computation models and their solution methods developed in thisapplication are for electrical power network. However, similarcomputation models and their solution methods can be developed usingtechniques developed in this application for all possible areas of studyand application that requires solution of linear or non-linearsimultaneous algebraic equations. Computation models and their solutionmethods could be for a system, a circuit, a machine, an apparatus, adevice, a material etc.It should be noted that ZPL or SZPL are embarrassingly parallelproblems, and readily amenable to parallel processing. This inventorbelieves, an approach outlined in the above is likely to work.If it works subject to verification by this inventor, it can producegrand simplifications in the sense that no need for specializedtriangulation and back-substitution or factorization software, and noneed for storing indexing and addressing information required forprocessing elements of factorized matrix. It appears the next numericalwonder is brewing. This inventor is humbled listening words Guardians ofGalaxy chanting: “Mr. Patel, You are the one, chosen”.The model constitutes {(69) or (70)}, {(71) or (72)} and {(73s) or(74s)}. It involves one solution of {(73) or (74)}. However, {(73a) or(73(c) or (74a) or (74c)} constitutes one equation for each node exceptthe Slack-node, and equations for all the nodes can be solved inparallel, just like Gauss numerical method with self-iteration foreach-node to handle local non-linearity. Self-iterations were introducedby this inventor first-time in the year 2005 in his patent # U.S. Pat.No. 7,788,051 and Canadian Patent #2548096 issued Jan. 5, 2011. This isa grand Gaussification of all the possible classical numerical methods.Equation {(73b) or (73d) or (74b) or (74d)} constitute one equation foreach node except the slack-node, and equations for all the nodes can besolved in sequence like Gauss-Seidel numerical method withself-iteration for each-node to handle local non-linearity. This is agrand Gauss-seidelization of all the possible classical numericalmethods. Gauss numerical method is ambarrassingly parallel. However, thebest approach seems to solve nodal equation for each node and nodalequations of its directly connected nodes in sequence like Gauss-Seidelnumerical method with self-iteration for each-node to handle localnon-linearity on separate processor simultaneously in parallel by thetechnique introduced by this inventor in his patent # U.S. Pat. No.7,788,051 and Canadian Patent #2548096 issued Jan. 5, 2011. Theparallelizaton technique of patent # U.S. Pat. No. 7,788,051 hasproduced 10-times speed-up in Y_(bus) formulation of Gauss-Seidelloadflow method involving self-iterations. The same parallelizationtechnique applied to models (69) and (70), could potentially produce20-to-40 times speed-ups. It looks like a revolution (Patelution) innumerical computation. The complex matrix [Z] in equations (69) and (70)can also be created by using building algorithm to create complexinverted coefficient matrix [C] or complex inverted admittance matrix[Y] and its different variations.

[V]=[Z] {[ΔI] or [I]} OR  (69)

[ΔV]=[Z][ΔI]  (70)

Wherein, though it is possible to write equations (69) and (70) incomplex form or real form in terms of real and imaginary components, ofinvolved variables/parameters relevant to problem being solved,development in the following is given only for complex versions ofequations (69) and (70) involving variables/parameter (voltage, current,and admittance) relevant to an electrical circuit or a network where,components of vectors [V], [I], [ΔV], [ΔI], and special Symbols aredefined in the following:

-   ^(q→p): means node q is directly connected to node-p-   _(q<p): means node-q among directly connected are processed prior to    the current node-p-   _(q>p): means node-q among directly connected are yet to be    processed after the current node-p-   nq: No. of off-diagonal elements in a row-p of [Z] that correspond    to directly connected nodes to a node-p-   nk: No. of off-diagonal elements in a row-p of [Z] that correspond    to not directly connected nodes to a node-p=(n−1)−nq-   n: No. of total elements in a row-p of [Z] that corresponds to total    no. of nodes or equations

$\begin{matrix}{{ZK}_{p} = {{\left\{ {{\sum\limits_{k = 1}^{p - 1}Z_{pk}} + {\sum\limits_{k = {p + 1}}^{n}Z_{pk}}} \right\}/n} - 1}} & \left( {71a} \right) \\{{{{IK}_{p} = {{\left\{ {{\sum\limits_{k = 1}^{p - 1}I_{k}} + {\sum\limits_{k = {p + 1}}^{n}I_{k}}} \right\}/\left( {n - 1} \right)}\mspace{14mu} {OR}}}\mspace{14mu} {{\Delta \; {IK}_{p}} = {\left\{ {{\sum\limits_{k = 1}^{p - 1}{\Delta \; I_{k}}} + {\sum\limits_{k = {p + 1}}^{n}{\Delta \; I_{k}}}} \right\}/\left( {n - 1} \right)}}}\mspace{11mu}} & \left( {71b} \right) \\{{{ZK}_{p} = {\left\{ {{\sum\limits_{\underset{k \neq q}{k = 1}}^{p - 1}Z_{pk}} + {\underset{k \neq q}{\sum\limits_{k = {p + 1}}^{n}}Z_{pk}}} \right\}/({nk})}}\mspace{11mu}} & \left( {71c} \right) \\{{{IK}_{p} = {{\left\{ {{\sum\limits_{\underset{k \neq q}{k = 1}}^{p - 1}I_{k}} + {\underset{k \neq q}{\sum\limits_{k = {p + 1}}^{n}}I_{k}}} \right\}/({nk})}\mspace{20mu} {OR}}}{{\Delta \; {IK}_{p}} = {\left\{ {{\sum\limits_{\underset{k \neq q}{k = 1}}^{p - 1}{\Delta \; I_{k}}} + {\underset{k \neq q}{\sum\limits_{k = {p + 1}}^{n}}I_{k}}} \right\}/({nk})}}} & \left( {71d} \right) \\{I_{p} = {{{SSH}_{p}^{*}/V_{p}^{*}} = {\left( {{PSH}_{p} - {jQSH}_{p}} \right)/\left( {e_{p} - {jf}_{p}} \right)}}} & \left( {72a} \right) \\{{\Delta \; I_{p}} = {{{SSH}_{p}^{*}/V_{p}^{*}} - {\left( {Y_{pp} + y_{p}} \right)V_{p}} - {\sum\limits_{q\rightarrow p}{Y_{pq}V_{q}}}}} & \left( {72b} \right)\end{matrix}$

Sparse Complex Matrix-Z Formulation:

$\begin{matrix}{V_{p} = {{Z_{pp}I_{p}} + {\sum\limits_{q\rightarrow p}{Z_{pq}I_{q}}}}} & \left( {73a} \right) \\\begin{matrix}{\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( I_{p} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\left( {n - 1} \right)\left( {ZK}_{p} \right)\left( {IK}_{p} \right)^{(r)}\text{:}}}} & {{{from}\mspace{14mu} \left( {71a} \right)},\left( {71b} \right)} \\{\left. {\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( I_{p} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q\rightarrow p}{Z_{pq}\left( I_{q} \right)}^{(r)}}}} \right) + {({nk})\left( {ZK}_{p} \right)\left( {IK}_{p} \right)^{(r)}\text{:}}} & {{{from}\mspace{14mu} \left( {71c} \right)},\left( {71d} \right)}\end{matrix} & \begin{matrix}\left( {73b} \right) \\\left( {73c} \right)\end{matrix} \\{\left. {\left. \left. {\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {Z_{pp}\left\lbrack I_{p} \right)}^{({sr})}} \right\}^{(r)} \right\rbrack + \left\lbrack {\sum\limits_{q\rightarrow p}^{q < p}{Z_{pq}\left( I_{q} \right)}^{({r + 1})}} \right) + {\sum\limits_{q\rightarrow p}^{q < p}{Z_{pq}\left( I_{q} \right)}^{(r)}}} \right\rbrack + {({nk})\left( {ZK}_{p} \right)\left( {IK}_{p} \right)^{(r)}\text{:}}} & \left( {73d} \right)\end{matrix}$

Full Complex Matrix-Z Formulation:

$\begin{matrix}{\mspace{79mu} {V_{p} = {{\sum\limits_{q = 1}^{n}{Z_{pq}I_{q}}} = {\sum\limits_{q = 1}^{n}{Z_{pq}\left( {{SSH}_{q}^{*}/V_{q}^{*}} \right)}}}}} & \left( {73e} \right) \\{\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack {{SSH}_{p}^{*}/\left\{ \left( V_{p}^{*} \right)^{({sr})} \right\}^{(r)}} \right\rbrack} + {\sum\limits_{q = 1}^{p - 1}{Z_{pq}\left( {{SSH}_{q}^{*}/\left( V_{q}^{*} \right)^{(r)}} \right)}} + {\sum\limits_{q = {p + 1}}^{n}{Z_{pq}\left( {{SSH}_{q}^{*}/\left( V_{q}^{*} \right)^{(r)}} \right)}}}} & \left( {73f} \right) \\{\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack {{SSH}_{p}^{*}/\left\{ \left( V_{p}^{*} \right)^{({sr})} \right\}^{(r)}} \right\rbrack} + \left\{ {{\sum\limits_{q = 1}^{p - 1}{Z_{pq}\left( {{SSH}_{q}^{*}/\left( V_{q}^{*} \right)^{({r + 1})}} \right)}} + {\sum\limits_{q = {p + 1}}^{n}{Z_{pq}\left( {{SSH}_{q}^{*}/\left( V_{q}^{*} \right)^{(r)}} \right)}}} \right\}}} & \left( {73g} \right)\end{matrix}$

Sparse Complex Matrix-Z Formulation:

$\begin{matrix}{\mspace{79mu} {{\Delta \; V_{p}} = {{Z_{pp}\Delta \; I_{p}} + {\sum\limits_{q->p}\; {Z_{pq}\Delta \; I_{q}}}}}} & \left( {74a} \right) \\{{\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\left( {n - 1} \right)\left( {ZK}_{p} \right)\left( {\Delta \; {IK}_{p}} \right)^{(r)}}}:{{from}\mspace{14mu} \left( {71a} \right)}}},\left( {71b} \right)} & \left( {74b} \right) \\{{\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q->p}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}} + {({nk})\left( {ZK}_{p} \right)\left( {\Delta \; {IK}_{p}} \right)^{(r)}}}:{{from}\mspace{14mu} \left( {71c} \right)}}},\left( {71d} \right)} & \left( {74c} \right) \\{\left. {\left. {\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q\rightarrow p}^{q > p}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{({r + 1})}}}} \right) + {\sum\limits_{q\rightarrow p}^{q > p}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}}} \right) + {({nk})\left( {ZK}_{p} \right)\left( {\Delta \; {IK}_{p}} \right)^{(r)}}} & \left( {74d} \right)\end{matrix}$

Full Complex Matrix-Z Formulation:

$\begin{matrix}{\mspace{85mu} {{\Delta \; V_{p}} = {\sum\limits_{q = 1}^{n}\; {Z_{pq}\Delta \; I_{q}}}}} & \left( {74e} \right) \\\left. {\left. {\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q = 1}^{p - 1}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}}}} \right) + {\sum\limits_{q = {p + 1}}^{n}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}}} \right) & \left( {74f} \right) \\\left. {\left. {\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q = 1}^{p - 1}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{({r + 1})}}}} \right) + {\sum\limits_{q = {p + 1}}^{n}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}}} \right) & \left( {74g} \right) \\{\mspace{79mu} {{{V_{p}^{({r + 1})} - V_{p}^{(r)}}} \leq ɛ}} & (75)\end{matrix}$

Matrix [Z] can also be made-up of real or complex components, and it isan inverse of coefficient matrix of linear and non-linear equationsorganized in different possible ways including in super-decoupled form,or an inverse of the Jacobian [J]⁻¹ or its different constant orapproximated variations including decoupled or super decoupled versions.It should be noted that equations (72a) and (72b) are the same as (60a)and (60) respectively, and (60s) and (63s) are different variations of(60).

The steps of loadflow calculation by ZPL or SZPL method are shown in theflowchart of FIG. 5. It should be noted that FIG. 5 and correspondingcalculation steps in the following are for complex inverted matrixbased, which is Gauss method without immediate updating as inGaus-Seidel method. Referring to the flowchart of FIG. 5, differentsteps are elaborated in steps marked with similar numbers in thefollowing. Four numbered steps are the inventive steps. The words “Readsystem data” in Step-a correspond to step-10 and step-20 in FIG. 7, andstep-16, step-18, step-24, step-36, step-38 in FIG. 8. All other stepsin the following correspond to step-30 in FIG. 7, and step-42, step-44,and step-46 in FIG. 8.

-   1. Read system data and assign an initial approximate solution    vector [V0], and store it. If better solution estimate is not    available, set voltage magnitude to 1.0 pu at load nodes and    specified values at PV-nodes, and angle of all nodes equal to that    of the slack-node, referred to as the flat-start.-   2. Form nodal impedance matrix [Y], and Initialize iteration count    ITR=0.-   3333. Form and store (m+k)×(m+k) size constant matrix [Z] of {(69)    or (70)}, using an algorithm or by inverting a coefficient matrix    [C] or the Jacobian matrix [J] or its different variants.-   4444. Compute the vector of {[I] or [ΔI]} using equations {(72a) or    (72b)}. However, use calculated value Q_(p) instead of QSH_(p) for    PV-nodes, and implement violated Q_(max) or Q_(min) limit of PV-node    generators.-   5555. Solve {(69) for [ΔV]} or {(70) for [V]}, and perform    Self-iterations for each node using {(73s) or (74s)}. It is also    possible to solve ((69) or (70)}, using Gauss method involving    self-iterations as per (73b) or (73c) or (73f) or (74b) or (74c), or    (74f), or using Gaus-Seidel method involving self-iterations as per    (73d) or (73g) or (74d) or (74g).-   6. Take a difference of vectors {[V]−[V0]}, find the maximum of the    element differences and store it in variable ‘MaxΔ ’, and perform    [V0]=[V] and ITR=ITR+1.-   8. Is MaxΔ less than specified tolerance? If it is not, Proceed to    step-4444 or else follow the next step.-   9. From calculated values of real and imaginary components of    complex voltage at PQ-nodes, real and imaginary components of    complex voltage and reactive power generation at PV-nodes, and tap    position of tap changing transformers, calculate power flows through    power network components.

Gauss-Seidel-Patel Loadflow (Gspl)

The complex conjugate power injected into the node-p of a power networkis given by the following equation (76) and its other alternativeorganizations.

$\begin{matrix}{{P_{p} - {jQ}_{p}} = {{V_{p}^{*}{\sum\limits_{q = 1}^{n}{Y_{pq}V_{q}}}} = {{{V_{p}^{*}\left( {Y_{pp} + y_{p}} \right)}V_{p}} + {V_{p}^{*}{\sum\limits_{q > p}\; {Y_{pq}V_{q}}}}}}} & (76) \\{{{\left( {{PSH}_{p} - {jQSH}_{p}} \right)/V_{p}^{*}} - {L_{p}V_{p}}} = {{\left( {Y_{pp} + y_{p}} \right)V_{p}} - {L_{p}V_{p}} + {\sum\limits_{q > p}\; {Y_{pq}V_{q}}}}} & (76) \\{\mspace{79mu} {{\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - {L_{p}V_{p}}} = {{\left( {Y_{pp} + y_{p} - L_{p}} \right)V_{p}} + {\sum\limits_{q > p}\; {Y_{pq}V_{q}}}}}} & (76) \\{V_{p} = {\left( {\sum\limits_{q > p}{Y_{pq}V_{q}}} \right)/\left\lbrack {\left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack}} & (76) \\{V_{p} = {\left\lbrack {\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - {L_{p}V_{p}} - {\sum\limits_{q > p}{Y_{pq}V_{q}}}} \right\rbrack/\left( {Y_{pp} + y_{p} - L_{p}} \right)}} & (76) \\{{\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} \right)} = {{\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{{SSH}_{p}^{*}/V_{s}^{2}}} \right)} \right\rbrack V_{p}} + {\sum\limits_{q > p}\; {Y_{pq}V_{q}}}}} & (76) \\{{\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} \right) - {\sum\limits_{q > p}\; {Y_{pq}V_{q}}}} = {\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{{SSH}_{p}^{*}/V_{s}^{2}}} \right)} \right\rbrack V_{p}}} & (76) \\{V_{p} = {\left\lbrack {\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} \right) - {\sum\limits_{q > p}{Y_{pq}V_{q}}}} \right\rbrack/\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{{SSH}_{p}^{*}/V_{s}^{2}}} \right)} \right\rbrack}} & (76) \\{\mspace{79mu} {{Where},}} & \; \\{\mspace{79mu} {{L_{p} = {- \infty}},\ldots \mspace{14mu},{- 1},0,{+ 1},\ldots \mspace{14mu},{{+ \infty}\mspace{14mu} \left( {{including}\mspace{14mu} {fractions}} \right)}}} & (77) \\{\mspace{79mu} {P_{p} = {{Re}\mspace{14mu} \left\{ {V_{p}^{*}{\sum\limits_{q = 1}^{n}\; {Y_{pq}V_{q}}}} \right\}}}} & (78) \\{\mspace{79mu} {Q_{p} = {{- {Im}}\left\{ {V_{p}^{*}{\sum\limits_{q = 1}^{n}\; {Y_{pq}V_{q}}}} \right\}}}} & (79)\end{matrix}$

Where, Re means “real part of” and Im means “imaginary part of”. Theequation (76) can also be written for complex power injected into thenode-p, instead of complex conjugate power injected into the node-p forthe purpose of the following development of a Gauss-Seidel-Patelnumerical and Loadflow method. However, detailed generalized propoundingstatement of the Gauss-Seidel-Patel numerical method will be provided inthe proposed book writing project.

The Gauss-Seidel-Patel (GSP) numerical method is for solving a set ofsimultaneous nonlinear algebraic equations iteratively. The GSPL-methodcalculates complex node voltage for any node-p as given in equation(76).

Iteration Process

Iterations start with the experienced/reasonable/logical guess for thesolution. The reference node also referred to as the slack-node voltagebeing specified, starting voltage guess is made for the remaining(n−1)-nodes in n-node network. Node voltage value is immediately updatedwith its newly calculated value in the iteration process in which onenode voltage is calculated at a time using latest updated other nodevoltage values. A node voltage value calculation at a time process isiterated over (n−1)-nodes in an n-node network, the reference nodevoltage being specified not required to be calculated.

Now, for the iteration-(r+1), the complex voltage calculation at node-pequation (76) and reactive power calculation at node-p equation (79),becomes:

$\begin{matrix}{V_{p}^{({r + 1})} = {\left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}{Y_{pq}V_{q}^{r}}}} \right)/\left\lbrack {\left\{ {\left( {{PSH}_{p} - {jQSH}_{p}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)^{r}} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack}} & (80) \\{V_{p}^{({r + 1})} = {\left\lbrack {\left( {{SSH}_{p}^{*}/\left( V_{p}^{*} \right)^{r}} \right) - {L_{p}V_{p}^{r}} - \left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}{Y_{pq}V_{q}^{r}}}} \right)} \right\rbrack/\left( {Y_{pp} + y_{p} - L_{p}} \right)}} & (80) \\{V_{p}^{({r + 1})} = {\left\lbrack {\left( {{SSH}_{p}^{*}/\left( V_{p}^{*} \right)^{r}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}^{r}/V_{s}^{2}}} \right) - \left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}{Y_{pq}V_{q}^{r}}}} \right)} \right\rbrack/\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{{SSH}_{p}^{*}/V_{s}^{2}}} \right)} \right\rbrack}} & (80) \\{Q_{p}^{({r + 1})} = {{- {Im}}\left\{ {{\left( V_{p}^{*} \right)^{r}{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}}} + {\left( V_{p}^{*} \right)^{r}{\sum\limits_{q = p}^{n}{Y_{pq}V_{q}^{r}}}}} \right\}}} & (81)\end{matrix}$

The well-known limitation of the Gauss-Seidel numerical method to be notable to converge to the high accuracy solution, was resolved by theintroduction of the concept of self-iteration of each calculatedvariable until convergence before proceeding to calculate the next. Thisis achieved by replacing equation (80) by equation (82) stated in thefollowing where self-iteration-(sr+1) over a node variable itself withinthe global iteration-(r+1) over (n−1) nodes in the n-node network isdepicted. During the self-iteration process only V_(p) and its real andimaginary components change without affecting any of the terms involvingV_(q). At the start of the self-iteration V_(p) ^(sr)=V_(p) ^(r), and atthe convergence of the self-iteration V_(p) ^((r+1))=V_(p) ^((sr+1))

$\begin{matrix}{\left( V_{p}^{({{sr} + 1})} \right)^{({r + 1})} = {\left( {{\sum\limits_{q = 1}^{p - 1}\; {Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}\; {Y_{pq}V_{q}^{r}}}} \right)/\left\lbrack {\left\{ {\left( {{PSH}_{p} - {jQSH}_{p}} \right)/\left( \left( {e_{p}^{2} + f_{p}^{2}} \right)^{sr} \right)^{r}} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack}} & (82) \\{\left. {\left. {\left( V_{p}^{({{sr} + 1})} \right)^{({r + 1})} = \left\lbrack {\left( {{SSH}_{p}^{*}/\left( V_{p}^{*} \right)^{sr}} \right)^{r} - {L_{p}\left( V_{p}^{*} \right)}^{sr}} \right)^{r}} \right) - \left( {{\sum\limits_{q = 1}^{p - 1}\; {Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}\; {Y_{pq}V_{q}^{r}}}} \right)} \right\rbrack/\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{{SSH}_{p}^{*}/V_{s}^{2}}} \right)} \right\rbrack} & (82) \\{\left. {\left. {\left( V_{p}^{({{sr} + 1})} \right)^{({r + 1})} = {\left\lbrack \left( {{SSH}_{p}^{*}/\left( V_{p}^{*} \right)^{sr}} \right)^{r} \right) - {\left( {L_{p}{{SSH}_{p}^{*}\left( V_{p} \right)}^{sr}} \right)^{r}/V_{s}^{2}}}} \right) - \left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}\; {Y_{pq}V_{q}^{r}}}} \right)} \right\rbrack/\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{{SSH}_{p}^{*}/V_{s}^{2}}} \right)} \right\rbrack} & (82)\end{matrix}$

Self-Convergence

The self-iteration process for a node is carried out until changes inthe real and imaginary parts of the node-p voltage calculated in twoconsecutive self-iterations are less than the specified tolerance. Ithas been possible to establish a relationship between the tolerancespecification for self-convergence and the tolerance specification forglobal-convergence. It is found sufficient for the self-convergencetolerance specification to be ten times the global-convergence tolerancespecification.

|Δf _(p) ^((sr+1)) =|f _(p) ^((sr+1)) −f _(p) ^(sr)|<10ε  (83)

|Δe _(p) ^((sr+1)) |=|e _(p) ^((sr+1)) −e _(p) ^(sr)|<10ε  (84)

For the global-convergence tolerance specification of 0.000001, it hasbeen found sufficient to have the self-convergence tolerancespecification of 0.00001 in order to have the maximum real and reactivepower mismatches of 0.0001 in the converged solution. However, for smallnetworks under not difficult to solve conditions they respectively couldbe 0.00001 and 0.0001 or 0.000001 and 0.0001, and for large networksunder difficult to solve conditions they sometimes need to berespectively 0.0000001 and 0.000001.

Convergence

The iteration process is carried out until changes in the real andimaginary parts of the set of (n−1)-node voltages calculated in twoconsecutive iterations are all less than the specified tolerance −ε, asshown in equations (85) and (86). The lower the value of the specifiedtolerance for convergence check, the greater the solution accuracy.

|Δf _(p) ^((r+1)) |=|f _(p) ^((r+1)) −f _(p) ^(r)|<ε  (85)

|Δe _(p) ^((r+1)) =|e _(p) ^((r+1)) −e _(p) ^(r)|<ε  (86)

Accelerated Convergence

The GSP-method being inherently slow to converge, it is characterized bythe use of an acceleration factor applied to the difference incalculated node voltage between two consecutive iterations to speed-upthe iterative solution process. The accelerated value of node-p voltageat iteration-(r+1) is given by

V _(p) ^((r+1))(accelerated)=V _(p) ^(r)+β(V _(p) ^((r+1)) −V _(p)^(r))  (87)

Where, β is the real number called acceleration factor, the value ofwhich for the best possible convergence for any given network can bedetermined by trial solutions. The GSP-method is very sensitive to thechoice of β, causing very slow convergence and even divergence for thewrong choice.

Scheduled or Specified Voltage at a PV-Node

Of the four variables, real power PSH_(p) and voltage magnitude VSH_(p)are scheduled/specified/set at a PV-node. If the reactive powercalculated using VSH_(p) at the PV-node is within the upper and lowergeneration capability limits of a PV-node generator, it is capable ofholding the specified voltage at its terminal. Therefore the complexvoltage calculated by equation (80) or (82) by using actually calculatedreactive power Q_(p) in place of QSH_(p) is adjusted to specifiedvoltage magnitude by equation (88). However, in case of violation ofupper or lower generation capability limits of a PV-node generator, aviolated limit value is used for QSH_(p) in (80) and (82), meaning aPV-node generator is no longer capable of holding its terminal voltageat its scheduled voltage magnitude VSH_(p), and the PV-node is switchedto a PQ-node type.

V _(p) ^((r+1))=(VSH _(p) V _(p) ^((r+1)))/|V _(p) ^((r+1))|  (88)

Calculation Steps of Gauss-Seidel-Patel Loadflow (GSPL) Method

The steps of loadflow calculation by GSPL method are shown in theflowchart of FIG. 6. Referring to the flowchart of FIG. 6, differentsteps are elaborated in steps marked with similar numbers in thefollowing. Steps marked with double numerals are the inventive steps.The words The words “Read system data” in Step-a correspond to step-10and step-20 in FIG. 7, and step-16, step-18, step-24, step-36, step-38in FIG. 8. All other steps in the following correspond to step-30 inFIG. 7, and step-42, step-44, and step-46 in FIG. 8.

-   71. Read system data and assign an initial approximate solution. If    better solution estimate is not available, set specified voltage    magnitude at PV-nodes, 1.0 p.u. voltage magnitude at PQ-nodes, and    all the node angles equal to that of the slack-node angle, which is    referred to as the flat-start.-   72. Form nodal admittance matrix, and Initialize iteration count r=1-   73. Scan all the node of a network, except the slack-node whose    voltage having been specified need not be calculated. Initialize    node count p=1, and initialize maximum change in real and imaginary    parts of node voltage variables DEMX=0.0 and DFMX=0.0-   74. Test for the type of a node at a time. For the slack-node go to    step-82, for a PQ-node go to the step-99, and for a PV-node follow    the next step.-   75. Compute Q_(p)(r+^(i)) for use as an imaginary part in    determining complex schedule power at a PV-node from equation (81)    after adjusting its complex voltage for specified value by equation    (88)-   76. If Q_(p) ^((r+1)) is greater than the upper reactive power    generation capability limit of the PV-node generator, set    QSH_(p)=the upper limit Q_(p) ^(max) for use in equation (82), and    go to step-99. If not, follow the next step.-   77. If Q_(p) ^((r+1)) is less than the lower reactive power    generation capability limit of the PV-node generator, set    QSH_(p)=the lower limit Q_(p) ^(min) for use in equation (82), and    go to step-99. If not, follow the next step.-   88. Compute V_(p) ^((r+1)) by equations (82), (83), (84) involving    self-iterations using QSH_(p)=Q_(p) ^((r+1)), and adjust for    specified voltage at the PV-node by equation (88), and go to    step-80.-   99. Compute V_(p) ^((r+1)) by equations (82), (83), (84) involving    self iteration-   80. Compute changes in the imaginary and real parts of the node-p    voltage by using equations (85) and (86), and replace current value    of DFMX and DEMX respectively in case any of them is larger.-   81. Calculate accelerated value of V_(p) ^((r+1)) by using equation    (87), and update voltage by V_(p) ^(r)=V_(p) ^((r+1)) for immediate    use in the next node voltage calculation.-   82. Check for if the total numbers of nodes−n are scanned. That is    if p<n, increment p=p+1, and go to step-74. Otherwise follow the    next step.-   83. If DEMX and DFMX both are not less than the convergence    tolerance (c) specified for the purpose of the accuracy of the    solution, advance iteration count r=r+1 and go to step-73, otherwise    follow the next step-   84. From calculated and known values of complex voltage at different    power network nodes, and tap position of tap changing transformers,    calculate power flows through power network components, and reactive    power generation at PV-nodes.

Patel Loadflow (PL)Model

Equations (3) and (4) can be organized in matrix form as per PatelNumerical Method:

$\begin{matrix}{\begin{pmatrix}{IR} \\{II}\end{pmatrix} = {\begin{pmatrix}{- B} & G \\{- G} & {- B}\end{pmatrix}\begin{pmatrix}f \\e\end{pmatrix}}} & (89)\end{matrix}$

Patel Transformation Decoupled Loadflow Model

[IR′]=[−Y][f]  (90)

[II′]=[−Y][e]  (91)

where,

IR _(p)′=(e _(p) PSH _(p) ′+f _(p) QSH _(p)′)/(e _(p) ² +f _(p) ²)  (92)

II _(p)′=(e _(p) QSH _(p) ′−f _(p) PSH _(p)′)/(e _(p) ² +f _(p) ²)  (93)

This is the model where elements of equations (90) and (91) are definedby following equations.

[−Y]=[−B]+[G][−B]⁻¹[G]  (94)

[IR′]=[IR]−[G][−B]⁻¹[II]  (95)

[II′]=[II]+[G][−B]⁻¹[RI]  (96)

Regular loadflow models can also be obtained by differentiating on bothsides of equations (89), (90) and (91).

Generalized Gauss-Seidel-Patel Numerical Method for Solution of Systemof Simultaneous Algebraic Equations Both Linear and Nonlinear:

A linear system of equations Ax=b can be written for any equation-p asequations (98) and (97). They can also be written in alternative formslike equation (76) including factor L_(p) of (77).

$\begin{matrix}{x_{p}^{({r + 1})} = {\left( {{\sum\limits_{q = 1}^{p - 1}\; {a_{pq}x_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}\; {a_{pq}x_{q}^{r}}}} \right)/\left\lbrack {\left\{ {b_{p}/\left( x_{p} \right)^{r}} \right\} - a_{pp}} \right\rbrack}} & (97) \\{\left( x_{p}^{({{sr} + 1})} \right)^{({r + 1})} = {\left( {{\sum\limits_{q = 1}^{p - 1}\; {a_{pq}x_{q}^{({r + 1})}}} - {\sum\limits_{q = {p + 1}}^{n}\; {a_{pq}x_{q}^{r}}}} \right)/\left\lbrack {\left\{ {b_{p}/\left( \left( x_{p} \right)^{sr} \right)^{r}} \right\} - a_{pp}} \right\rbrack}} & (98)\end{matrix}$

A nonlinear system of equations f(x)=y can be written for any equation-pas equations (82), which is specifically a nonlinear power flow equationof a power network involving complex variables and constant parameters.

Equations (98) and (82) are defining equations of GeneralizedGauss-Seidel-Patel numerical method involving self-iterations. It shouldbe noted that self-iterations within global iterations are analogous tothe earth rotating on its own axis while making rounds around the Sun.This generalized approach for solution of both linear and nonlinearsystem of simultaneous algebraic equations could potentially be amenableto acceleration factors greater than 2 unlike original Gauss-Seidelnumerical method subject to experimental numerical verification. Furtherverbal elaborations about the Generalized Gauss-Seidel-Patel numericalmethod will be provided as part of the proposed book writing project.

GENERAL STATEMENTS

The system stores a representation of the reactive capabilitycharacteristic of each machine and these characteristics act asconstraints on the reactive power, which can be calculated for eachmachine.

While the description above refers to particular embodiments of thepresent invention, it will be understood that many modifications may bemade without departing from the spirit thereof. The accompanying claimsare intended to cover such modifications as would fall within the truescope and spirit of the present invention.

The presently disclosed embodiments are therefore to be considered inall respect as illustrative and not restrictive, the scope of theinvention being indicated by the appended claims in addition to theforegoing description, and all changes which come within the meaning andrange of equivalency of the claims are therefore intended to be embracedtherein.

What is claimed is:
 1. A Method of forming and solving a Loadflowcomputation model of a power network to affect control of voltages andpower flows in a power system, comprising the steps of: obtainingon-line or simulated data of open or close status of all switches andcircuit breakers in the power network, and reading data of operatinglimits of components of the power network including maximumVoltage×Ampere (VA or MVA) carrying capability limits of transmissionlines, transformers, and PV-node, a generator-node where Real-Power-Pand Voltage-Magnitude-V are specified, maximum and minimum reactivepower generation capability limits of generators, and transformers tapposition limits, obtaining on-line readings of specified Real-Power-Pand Reactive-Power-Q at PQ-nodes, Real-Power-P and voltage-magnitude-Vat PV-nodes, voltage magnitude and angle at a slack node, andtransformer turns ratios, wherein said on-line readings are thecontrolled variables, performing loadflow computation by forming andsolving a loadflow computation model of the power network to calculate,complex voltages or their real and imaginary components or voltagemagnitude and voltage angle at nodes of the power network providing forcalculation of power flow through different components of the powernetwork, and to calculate reactive power generations at PV-nodes andslack node, real power generation at the slack node and transformertap-position indications of tap-changing transformers in dependence ofthe said obtained on-line readings of given or specified values of thecontrolled variables or parameters and physical limits of operation ofthe power network components, the said loadflow computation model of thepower network is referred to as a Patel Super Decoupled Loadflow(PSDL-YY2) computation model characterized by and comprises equations{(32) to (35)} or {(36) to (37)}, {(3), (4), (51), (52), (40c), and(53b)}, or {(42), (43), (15), (16), (40), and (53a)}, (39), and {(54)and (55)} given below:[Δf]=[Yf]⁻¹[ΔRI′]  (32)[f]=[f]+[Δf]  (33)[Δe]=[Ye]⁻¹[ΔII′]  (34)[e]=[e]+[Δe]  (35)[f]=[Yf]⁻¹{[ΔRI′] or [RI′]}  (36)[e]=[Ye]⁻¹{[ΔII′] or [II′]}  (37) where, components of vectors [RI′],[ΔRI′], [II′], [ΔII′], and matrices [Yf], [Ye] are defined in thefollowing: $\begin{matrix}{{RI}_{p} = {{\left( {{e_{p}{PSH}_{p}} + {f_{p}{QSH}_{p}}} \right)\left( {e_{p}^{2} + f_{p}^{2}} \right)} = {{- \left\lbrack {{\left( {B_{pp} + b_{p}} \right)f_{p}} + {\sum\limits_{q > p}{B_{pq}f_{q}}}} \right\rbrack} + \left\lbrack {\left( {G_{pp} + g_{p}} \right)e_{p}{\underset{q > p}{+ \sum}{G_{pq}e_{q}}}} \right\rbrack}}} & (3) \\{{II}_{p} = {{\left( {{e_{p}{QSH}_{p}} + {f_{p}{PSH}_{p}}} \right)\left( {e_{p}^{2} + f_{p}^{2}} \right)} = {{- \left\lbrack {{\left( {G_{pp} + g_{p}} \right)f_{p}} + {\sum\limits_{q > p}{G_{pq}f_{q}}}} \right\rbrack} - \left\lbrack {{\left( {B_{pp} + b_{p}} \right)e_{p}} + {\sum\limits_{q > p}{B_{pq}e_{q}}}} \right\rbrack}}} & (4) \\{{\Delta \; {RI}_{p}} \approx {\left\lbrack {\left( {{e_{p}{PSH}_{p}} + {f_{p}{QSH}_{p}}} \right)/\left( {e_{s}^{2} + f_{s}^{2}} \right)} \right\rbrack - \left\lbrack {\left( {{e_{p}{PSH}_{p}} + {f_{p}{QSH}_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\rbrack}} & (15) \\{{\Delta \; {II}_{p}} \approx {\left\lbrack {\left( {{e_{p}{QSH}_{p}} + {f_{p}{PSH}_{p}}} \right)/\left( {e_{s}^{2} + f_{s}^{2}} \right)} \right\rbrack - \left\lbrack {\left( {{e_{p}{QSH}_{p}} + {f_{p}{PSH}_{p}}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\rbrack}} & (16) \\{\mspace{79mu} {{\Delta \; {RI}_{p}^{\prime}} = {{\Delta \; {RI}_{p}{Cos}\; \Phi_{p}} + {\Delta \; {II}_{p}{Sin}\; \Phi_{p}}}}} & (42) \\{\mspace{79mu} {{\Delta \; I\; I_{p}^{\prime}} = {{\Delta \; {II}_{p}{Cos}\; \Phi_{p}} + {\Delta \; {RI}_{p}{Sin}\; \Phi_{p}}}}} & (43) \\{\mspace{79mu} {{Yf}_{pp} = {{Ye}_{pp} = {b_{p}^{\prime} + {\sum\limits_{q > p}{- {Yf}_{pq}}}}}}} & (39) \\{\mspace{79mu} {b_{p}^{\prime} = {{\left( {{{QSH}_{p}{Cos}\; \Phi_{p}} - {{PSH}_{p}{Sin}\; \Phi_{p}}} \right)/\left( {e_{s}^{2} + f_{s}^{2}} \right)} + {b_{p}{Cos}\; \Phi_{p}}}}} & (40) \\{\mspace{79mu} {b_{p}^{\prime} = {{- b_{p}}{Cos}\; \Phi_{p}}}} & \left( {40c} \right) \\{\mspace{79mu} {{RI}_{p}^{\prime} = {{{RI}_{p}{Cos}\; \Phi_{p}} + {{II}_{p}{Sin}\; \Phi_{p}}}}} & (51) \\{\mspace{79mu} {{II}_{p}^{\prime} = {{{II}_{p}{Cos}\; \Phi_{p}} - {{RI}_{p}{Sin}\; \Phi_{p}}}}} & (52) \\{{Yf}_{pq} = {{Yf}_{pq} = \left\lbrack \begin{matrix}{Y_{pq}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} r\text{/}x\mspace{14mu} {ratio}} \leq 3.0} \\{\left( {B_{pq} + {0.9\left( {Y_{pq} - B_{pq}} \right)}} \right)\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} r\text{/}x\mspace{14mu} {ratio}} > 3.0}\end{matrix} \right.}} & \left( {53a} \right) \\{{Yf}_{pq} = {{Yf}_{pq} = \left\lbrack \begin{matrix}{{- Y_{pq}}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} r\text{/}x\mspace{14mu} {ratio}} \leq 3.0} \\{{- \left( {B_{pq} + {0.9\left( {Y_{pq} - B_{pq}} \right)}} \right)}\text{:}} & {{{for}\mspace{14mu} {branch}\mspace{14mu} r\text{/}x\mspace{14mu} {ratio}} > 3.0}\end{matrix} \right.}} & \left( {53b} \right) \\{\mspace{79mu} {\left\lbrack f_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = \left\lbrack \left\{ {\left( {{RI}_{p}^{\prime}\mspace{14mu} {or}\mspace{14mu} \Delta \; R\; I_{p}^{\prime}} \right)/{Yf}_{pp}} \right\}^{({sr})} \right\rbrack^{(r)}}} & (54) \\{\mspace{79mu} {\left\lbrack e_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = \left\lbrack \left\{ {\left( {{II}_{p}^{\prime}\mspace{14mu} {or}\mspace{14mu} \Delta \; I\; I_{p}^{\prime}} \right)/{Ye}_{pp}} \right\}^{({sr})} \right\rbrack^{(r)}}} & (55)\end{matrix}$ where, different symbols and terms are defined as follows:Y_(pq)=G_(pq)+jB_(pq): (p−q) th element of nodal admittance matrixwithout shunts Y_(pp)=G_(pp)+jB_(pp): p-th diagonal element of nodaladmittance matrix without shunts Y_(pq)=|Y_(pq)|=Sqrt(G_(pq) ²±B_(pq)²): magnitude of complex Y_(pq) y_(p)=g_(p)+jb_(p): total shuntadmittance at any node-p V_(p)=e_(p)+jf_(p)=V_(p)∠θ_(p): complex voltageof any node-p V_(s)=e_(s)+jf_(s)=V_(s)∠θ_(s): complex slack-node voltageΔf_(p), Δe_(p): imaginary, real part of complex voltage correctionsRI_(p)+jII_(p): net nodal injected current, calculated ΔRI_(p)+jΔII_(p):nodal injected current residue or mismatch SSH_(p)=PSH_(p)+jQSH_(p): netnodal injected power, scheduled/specified C_(p)=1∠Φ_(p)=Cos Φ_(p)+jSinΦ_(p): Unitary rotation/transformation sr: nodal self-iteration count r:global iteration count q>p: node-q is connected to node-p excluding thecase of q=p PQ-node: load-node, where, Real-Power-P and Reactive-Power-Qare specified PV-node: generator-node, where, Real-Power-P andVoltage-Magnitude-V are specified V_(s)≈V_(B)≈V_(N): slack-node voltagemagnitude, base value, and nominal value of voltage magnitude are veryclosely similar, and therefore, they can be used interchangeably,evaluating loadflow computation for any over loaded components of thepower network and for under or over voltage at any of the nodes of thepower network, correcting one or more controlled variables and repeatingthe performing loadflow computation, evaluating, and correcting stepsuntil evaluating step finds no over loaded components and no under orover voltages in the power network, and affecting a change in power flowthrough components of the power network and voltage magnitudes andangles at the nodes of the power network by actually implementing thefinally obtained values of controlled variables after evaluating stepfinds a good power system or stated alternatively the power networkwithout any overloaded components and under or over voltages, whichfinally obtained controlled variables however are stored for acting uponfast in case a simulated event actually occurs.
 2. A Method as definedin claim 1 wherein, the said loadflow computation model of the powernetwork is referred to as a complex matrix [C] based Patel Loadflow-2(CPL-2) model characterized by and comprises equations {(56) to (68)}listed in the following:[ΔI]=[C][ΔV]  (56)[ΔV]=[C]⁻¹[ΔI]  (57)OR{[ΔI] or [I]}=[C][V]  (58)[V]=[C]⁻¹{[ΔI] or [I]}  (59) where, components of complex vectors [I],[ΔI] and complex matrix [C] are defined in the following:$\begin{matrix}{I_{p} = {{\left( {{PSH}_{p} - {jQSH}_{p}} \right)/\left( {e_{p} - {jf}_{p}} \right)} = {\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) = \left\lbrack {{\left( {Y_{pp} + y_{p}} \right)V_{p}} + {\sum\limits_{q > p}{Y_{pq}V_{q}}}} \right\rbrack}}} & \left( {60a} \right) \\{{\left. {{\Delta \; I_{p}} = {{\left( {{SSH}_{p}^{*} - S_{p}^{*}} \right)/V_{p}^{*}} = {\left( {{PSH}_{p} - {jQSH}_{p}} \right) - \left( {P_{p} - {jQ}_{p}} \right)}}} \right\rbrack/V_{p}^{*}} = {\left( {{\Delta \; P_{p}} - {j\; \Delta \; Q_{p}}} \right)/V_{p}^{*}}} & (60) \\{\mspace{79mu} {{\Delta \; I_{p}} = {{\left\lbrack {\left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack V_{p}} - {\sum\limits_{q > p}{Y_{pq}V_{q}}}}}} & (60) \\{{{\Delta \; I_{p}} \approx {\left\lbrack {\left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}} \right\} - \left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\}} \right\rbrack V_{p}}} = {{L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{*}}} - {{SSH}_{p}^{*}/V_{p}^{*}}}} & (60) \\{\mspace{79mu} {{\Delta \; I_{p}} = {{\left\lbrack {L_{p} - \left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\}} \right\rbrack V_{p}} = {{L_{p}V_{p}} - {{SSH}_{p}^{*}/V_{p}^{*}}}}}} & (60) \\{\mspace{79mu} {C_{pq} = {- Y_{pq}}}} & (61) \\{C_{pp} = {\left\lbrack {\left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)}} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack \approx \left\lbrack {\left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack}} & (62) \\{\mspace{79mu} {C_{pp} = {\left\lbrack {L_{p} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack \mspace{14mu} {OR}}}} & (62) \\{{\Delta \; I_{p}} = {{\left( {S_{p}^{*} - {SSH}_{p}^{*}} \right)/V_{p}^{*}} = {{\left\lbrack {\left( {P_{p} - {jQ}_{p}} \right) - \left( {{PSH}_{p} - {jQSH}_{p}} \right)} \right\rbrack/V_{p}^{*}} = {\left\lbrack {\left( {{- \Delta}\; P_{p}} \right) - {j\left( {{- \Delta}\; Q_{p}} \right)}} \right\rbrack/V_{p}^{*}}}}} & (63) \\{\mspace{79mu} {{\Delta \; I_{p}} = {{\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\}} \right\rbrack V_{p}} + {\sum\limits_{q > p}{Y_{pq}V_{q}}}}}} & (63) \\{{{\Delta \; I_{p}} \approx {\left\lbrack {\left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\} - \left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}} \right\}} \right\rbrack V_{p}}} = {{{SSH}_{p}^{*}/V_{p}^{*}} - {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}}}} & (63) \\{\mspace{79mu} {{\Delta \; I_{p}} = {{\left\lbrack {\left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\} - L_{p}} \right\rbrack V_{p}} = {{{SSH}_{p}^{*}/V_{p}^{*}} - {L_{p}V_{p}}}}}} & (63) \\{\mspace{79mu} {C_{pq} = Y_{pq}}} & (64) \\{C_{pp} = {\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\}} \right\rbrack \approx \left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left\{ {L_{p}{{SSH}_{p}^{*}/\left( {e_{s}^{2} + f_{s}^{2}} \right)}} \right\}} \right\rbrack}} & (65) \\{\mspace{79mu} {C_{pp} = \left\lbrack {\left( {Y_{pp} + y_{p}} \right) - L_{p}} \right\rbrack}} & (65) \\{\mspace{79mu} {C_{pp} = \left( {Y_{pp} + y_{p}} \right)}} & \left( {65a} \right) \\{\mspace{79mu} {\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = \left\lbrack \left( {\Delta \; {I_{p}/C_{pp}}} \right)^{({sr})} \right\rbrack^{(r)}}} & (66) \\{\mspace{79mu} {\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = \left\lbrack \left( {\left( {\Delta \; I_{p}\mspace{14mu} {or}\mspace{14mu} I_{p}} \right)/C_{pp}} \right)^{({sr})} \right\rbrack^{(r)}}} & (67) \\{\mspace{79mu} {{L_{p} = {- \infty}},\ldots \mspace{14mu},{- 1},0,{+ 1},\ldots \mspace{14mu},{{+ \infty}\mspace{14mu} \left( {{including}\mspace{14mu} {fractions}} \right)}}} & (68)\end{matrix}$ where, different symbols and terms are defined as follows:Y_(pq)=G_(pq)+jB_(pq): (p−q) th element of nodal admittance matrixwithout shunts Y_(pp)=G_(pp)+jB_(pp): p-th diagonal element of nodaladmittance matrix without shunts y_(p)=g_(p)+jb_(p): total shuntadmittance at any node-p V_(p)=e_(p)+jf_(p)=V_(p)∠θ_(p): complex voltageof any node-p V_(s)=e_(s)+jf_(s)=V_(s)∠θ_(s): complex slack-node voltageΔV_(p)=Δe_(p)+jΔf_(p): complex voltage correctionsSSH_(p)=PSH_(p)+jQSH_(p): net nodal injected power, scheduled/specifiedsr: nodal self-iteration count r: global iteration count q>p: node-q isconnected to node-p excluding the case of q=p PQ-node: load-node, where,Real-Power-P and Reactive-Power-Q are specified PV-node: generator-node,where, Real-Power-P and Voltage-Magnitude-V are specifiedV_(s)≈V_(B)≈V_(N): slack-node voltage magnitude, base value, and nominalvalue of voltage magnitude are very closely similar, and therefore, theycan be used interchangeably.
 3. A Method as defined in claim 1 wherein,the said loadflow computation model of the power network is referred toas Gauss-Seidel-Patel Loadflow (GSPL) computation model characterized byand comprises equations (76) to (88) listed in the following:$\begin{matrix}{\mspace{79mu} {{P_{p} - {jQ}_{p}} = {{V_{p}^{*}{\sum\limits_{q = 1}^{n}\; {Y_{pq}V_{q}}}} = {{{V_{p}^{*}\left( {Y_{pp} + y_{p}} \right)}V_{p}} + {V_{p}^{*}{\sum\limits_{q > p}{Y_{pq}V_{q}}}}}}}} & (76) \\{{{\left( {{PSH}_{p} - {jQSH}_{p}} \right)/V_{p}^{*}} - {L_{p}V_{p}}} = {{\left( {Y_{pp} + y_{p}} \right)V_{p}} - {L_{p}V_{p}} + {\sum\limits_{q > p}{Y_{pq}V_{q}}}}} & (76) \\{\mspace{79mu} {{\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - {L_{p}V_{p}}} = {{\left( {Y_{pp} + y_{p} - L_{p}} \right)V_{p}} + {\sum\limits_{q > p}{Y_{pq}V_{q}}}}}} & (76) \\{V_{p} = {\left( {\sum\limits_{q > p}{Y_{pq}V_{q}}} \right)/\left\lbrack {\left\{ {{SSH}_{p}^{*}/\left( {e_{p}^{2} + f_{p}^{2}} \right)} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack}} & (76) \\{V_{p} = {\left\lbrack {\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - {L_{p}V_{p}} - {\sum\limits_{q > p}{Y_{pq}V_{q}}}} \right\rbrack/\left( {Y_{pp} + y_{p} - L_{p}} \right)}} & (76) \\{{\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} \right)} = {{\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} \right)} \right\rbrack V_{p}} + {\sum\limits_{q > p}{Y_{pq}V_{q}}}}} & (76) \\{{\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} \right) - {\sum\limits_{q > p}{Y_{pq}V_{q}}}} = {\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} \right)} \right\rbrack V_{p}}} & (76) \\{V_{p} = {\left\lbrack {\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} \right) - {\sum\limits_{q > p}{Y_{pq}V_{q}}}} \right\rbrack/\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}/V_{s}^{2}}} \right)} \right\rbrack}} & (76) \\{\mspace{79mu} {{where},}} & \; \\{\mspace{76mu} {{L_{p} = {- \infty}},\ldots \mspace{14mu},{- 1},0,{+ 1},\ldots \mspace{14mu},{{+ \infty}\mspace{14mu} \left( {{including}\mspace{14mu} {fractions}} \right)}}} & (77) \\{\mspace{76mu} {P_{p} = {{Re}\mspace{14mu} \left\{ {V_{p}^{*}{\sum\limits_{q = 1}^{n}\; {Y_{pq}V_{q}}}} \right\}}}} & (78) \\{\mspace{76mu} {Q_{p} = {{- {Im}}\left\{ {V_{p}^{*}{\sum\limits_{q = 1}^{n}\; {Y_{pq}V_{q}}}} \right\}}}} & (79) \\{V_{p}^{({r + 1})} = {\left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}{Y_{pq}V_{q}^{r}}}} \right)/\left\lbrack {\left\{ {\left( {{PSH}_{p} - {jQSH}_{p}} \right)/\left( {e_{p}^{2} + f_{p}^{2}} \right)^{r}} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack}} & (80) \\{\left. {V_{p}^{({r + 1})} = {\left\lbrack \left( {{SSH}_{p}^{*}/V_{p}^{*}} \right)^{r} \right) - {L_{p}V_{p}^{r}} - \left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}{Y_{pq}V_{q}^{r}}}} \right)}} \right\rbrack/\left( {Y_{pp} + y_{p} - L_{p}} \right)} & (80) \\{V_{p}^{({r + 1})} = \left\lbrack {\left( {{SSH}_{p}^{*}/\left( V_{p}^{*} \right)^{r}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}^{r}/V_{s}^{2}}} \right) - {\left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}{Y_{pq}V_{q}^{r}}}} \right\rbrack/\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}^{r}/V_{s}^{2}}} \right)} \right\rbrack}} \right.} & (80) \\{Q_{p}^{({r + 1})} = {{- {Im}}\mspace{14mu} \left\{ {{\left( V_{p}^{*} \right)^{r}{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}}} + {\left( V_{p}^{*} \right)^{r}{\sum\limits_{q = p}^{n}\; {Y_{pq}V_{q}^{r}}}}} \right\}}} & (81) \\{\left( V_{p}^{({{sr} + 1})} \right)^{({r + 1})} = {\left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}{Y_{pq}V_{q}^{r}}}} \right)/\left\lbrack {\left\{ {\left( {{PSH}_{p} - {jQSH}_{p}} \right)/\left( \left( {e_{p}^{2} + f_{p}^{2}} \right)^{sr} \right)^{r}} \right\} - \left( {Y_{pp} + y_{p}} \right)} \right\rbrack}} & (82) \\{\left. {\left( V_{p}^{({{sr} + 1})} \right)^{({r + 1})} = {\left\lbrack {\left( {{SSH}_{p}^{*}/\left( V_{p}^{*} \right)^{sr}} \right)^{r} - {L_{p}\left( V_{p} \right)}^{sr}} \right)^{r} - \left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}{Y_{pq}V_{q}^{r}}}} \right)}} \right\rbrack/\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}^{r}/V_{s}^{2}}} \right)} \right\rbrack} & (82) \\{{\left. {\left. \; {\left( V_{p}^{({{sr} + 1})} \right)^{({r + 1})} = {\left\lbrack \left( {{SSH}_{p}^{*}/\left( V_{p}^{*} \right)^{sr}} \right)^{r} \right) - {\left( {L_{p}{{SSH}_{p}^{*}\left( V_{p} \right)}^{sr}} \right)^{r}/V_{s}^{2}}}} \right) - \left( {{\sum\limits_{q = 1}^{p - 1}{Y_{pq}V_{q}^{({r + 1})}}} + {\sum\limits_{q = {p + 1}}^{n}{Y_{pq}V_{q}^{r}}}} \right)} \right\rbrack/\left\lbrack {\left( {Y_{pp} + y_{p}} \right) - \left( {L_{p}{SSH}_{p}^{*}{V_{p}^{r}/V_{s}^{2}}} \right)} \right\rbrack}\mspace{76mu}} & (82) \\{\mspace{76mu} {{{\Delta \; f_{p}^{({{sr} + 1})}}} = {{{f_{p}^{({{sr} + 1})} - f_{p}^{sr}}} < {10ɛ}}}\;} & (83) \\{\mspace{76mu} {{{\Delta \; e_{p}^{({r + 1})}}} = {{{e_{p}^{({r + 1})} - e_{p}^{sr}}} < {10ɛ}}}\mspace{76mu}} & (84) \\{\mspace{76mu} {{{\Delta \; f_{p}^{({r + 1})}}} = {{{f_{p}^{({r + 1})} - f_{p}^{r}}} < ɛ}}\mspace{76mu}} & (85) \\{\mspace{76mu} {{{\Delta \; e_{p}^{({r + 1})}}} = {{{e_{p}^{({r + 1})} - e_{p}^{r}}} < ɛ}}\mspace{50mu}} & (86) \\{\mspace{76mu} {{V_{p}^{({r + 1})}({accelerated})} = {V_{p}^{r} + {\beta \left( {V_{p}^{({r + 1})} - V_{p}^{r}} \right)}}}} & (87) \\{\mspace{76mu} {V_{p}^{({r + 1})} = {\left( {{VSH}_{p}V_{p}^{({r + 1})}} \right)/{V_{p}^{({r + 1})}}}}} & (88)\end{matrix}$ where, different symbols and terms are defined as follows:Y_(pq)=G_(pq)+jB_(pq): (p−q) th element of nodal admittance matrixwithout shunts Y_(pp)=G_(pp)+jB_(pp): p-th diagonal element of nodaladmittance matrix without shunts y_(p)=g_(p)+jb_(p): total shuntadmittance at any node-p V_(p)=e_(p)+jf_(p)=V_(p)∠θ_(p): complex voltageof any node-p V_(s)=e_(s)+jf_(s)=V_(s)∠θ_(s): complex slack-node voltageSSH_(p)=PSH_(p)+jQSH_(p): net nodal injected power, scheduled/specifiedβ: real acceleration factor sr: nodal self-iteration count r: networkwide global iteration count q>p: node-q is connected to node-p excludingthe case of q=p PQ-node: load-node, where, Real-Power-P andReactive-Power-Q are specified PV-node: generator-node, where,Real-Power-P and Voltage-Magnitude-V are specified V_(s)≈V_(B)≈V_(N):slack-node voltage magnitude, base value, and nominal value of voltagemagnitude are very closely similar, and therefore, they can be usedinterchangeably.
 4. A Method of forming and solving a Loadflowcomputation model of a power network to affect control of voltages andpower flows in a power system, comprising the steps of: obtainingon-line or simulated data of open or close status of all switches andcircuit breakers in the power network, and reading data of operatinglimits of components of the power network including maximum Voltage xAmpere (VA or MVA) carrying capability limits of transmission lines,transformers, and PV-node, a generator-node where Real-Power-P andVoltage-Magnitude-V are specified, maximum and minimum reactive powergeneration capability limits of generators, and transformers tapposition limits, obtaining on-line readings of specified Real-Power-Pand Reactive-Power-Q at PQ-nodes, Real-Power-P and voltage-magnitude-Vat PV-nodes, voltage magnitude and angle at a slack node, andtransformer turns ratios, wherein said on-line readings are thecontrolled variables, performing loadflow computation by forming andsolving a loadflow computation model of the power network to calculate,complex voltages or their real and imaginary components or voltagemagnitude and voltage angle at nodes of the power network providing forcalculation of power flow through different components of the powernetwork, and to calculate reactive power generations at PV-nodes andslack node, real power generation at the slack node and transformertap-position indications of tap-changing transformers in dependence ofthe said obtained on-line readings of given or specified values of thecontrolled variables or parameters and physical limits of operation ofthe power network components, the said loadflow model of the powernetwork referred to as a [C]⁻¹ or a Z-matrix based Patel Loadflow—(CIPLor ZPL) as well as its sparse version referred to as a SCIPL or a SZPLcharacterized by and comprises equations {(69) to (75)} listed in thefollowing:[V]=[Z]{[ΔI] or [I]} OR  (69)[ΔV]=[Z][ΔI]  (70) Wherein, though it is possible to write equations(69) and (70) in complex form or real form in terms of real andimaginary components, of involved variables/parameters relevant toproblem being solved, development in the following is given only forcomplex versions of equations (69) and (70) involvingvariables/parameter (voltage, current, and admittance) relevant to anelectrical circuit or a network where, components of vectors [V], [I],[ΔV], [ΔI], and special Symbols are defined in the following: ^(q→p):means node q is directly connected to node-p _(q<p): means node-q amongdirectly connected are processed prior to the current node-p _(q>p):means node-q among directly connected are yet to be processed after thecurrent node-p nq: No. of off-diagonal elements in a row-p of [Z] thatcorrespond to directly connected nodes to a node-p nk: No. ofoff-diagonal elements in a row-p of [Z] that correspond to not directlyconnected nodes to a node-p=(n−1)−nq n: No. of total elements in a row-pof [Z] that corresponds to total no. of nodes or equations$\begin{matrix}{{ZK}_{p} = {\left\{ {{\sum\limits_{k = 1}^{p - 1}\; Z_{p\; k}} + {\sum\limits_{k = {p + 1}}^{n}\; Z_{p\; k}}} \right\}/\left( {n - 1} \right)}} & \left( {71a} \right) \\{{{IK}_{p} = {{\left\{ {{\sum\limits_{k = 1}^{p - 1}\; I_{k}} + {\sum\limits_{k = {p + 1}}^{n}\; I_{k}}} \right\}/\left( {n - 1} \right)}\mspace{14mu} {OR}}}{{\Delta \; {IK}_{p}} = {\left\{ {{\sum\limits_{k = 1}^{p - 1}\; {\Delta \; I_{k}}} + {\sum\limits_{k = {p + 1}}^{n}\; {\Delta \; I_{k}}}} \right\}/\left( {n - 1} \right)}}} & \left( {71b} \right) \\{{ZK}_{p} = {\left\{ {{\sum\limits_{\underset{k \neq q}{k = 1}}^{p - 1}\; Z_{p\; k}} + \sum\limits_{\underset{k \neq q}{k = {p + 1}}}^{n}} \right\} \; {Z_{p\; k}/\left( {n\; k} \right)}}} & \left( {71c} \right) \\{{{IK}_{p} = {{\left\{ {{\sum\limits_{\underset{k \neq q}{k = 1}}^{p - 1}\; {\Delta \; I_{k}}} + {\sum\limits_{\underset{k \neq q}{k = {p + 1}}}^{n}I_{k}}} \right\} \;/\left( {n\; k} \right)}\mspace{14mu} {OR}}}{{\Delta \; {IK}_{p}} = {\left\{ {{\sum\limits_{\underset{k \neq q}{k = 1}}^{p - 1}\; {\Delta \; I_{k}}} + {\sum\limits_{\underset{k \neq q}{k = {p + 1}}}^{n}{\Delta \; I_{k}}}} \right\} \;/\left( {n\; k} \right)}}} & \left( {71d} \right) \\{I_{p} = {{{SSH}_{p}^{*}/V_{p}^{*}} = {\left( {{PSH}_{p} - {jQSH}_{p}} \right)/\left( {e_{p} - {jf}_{p}} \right)}}} & \left( {72a} \right) \\{{\Delta \; I_{p}} = {{{SSH}_{p}^{*}/V_{p}^{*}} - {\left( {Y_{pp} + y_{p}} \right)V_{p}} - {\sum\limits_{q->p}\; {Y_{pq}V_{q}}}}} & \left( {72b} \right)\end{matrix}$ Sparse Complex Matrix-Z Formulation: $\begin{matrix}{\mspace{79mu} {V_{p} = {{Z_{pp}I_{p}} + {\sum\limits_{q->p}\; {Z_{pq}I_{q}}}}}} & \left( {73a} \right) \\{{\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {{{Z_{pp}\left\lbrack \left\{ \left( I_{p} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\left( {n - 1} \right)\left( {ZK}_{p} \right)\left( {IK}_{p} \right)^{(r)}}}:{{from}\mspace{14mu} \left( {71a} \right)}}}, \left( {71b} \right)} & \left( {73b} \right) \\{\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( I_{p} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q->p}\; {Z_{pq}\left( I_{q} \right)}^{(r)}} + {\quad{{{({nk})\left( {ZK}_{p} \right)\left( {IK}_{p} \right)^{(r)}}:{{from}\mspace{14mu} \left( {71c} \right)}},\left( {71d} \right)}}}} & \left( {73c} \right) \\{\left. {\left. \left. {\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {Z_{pp}\left\lbrack I_{p} \right)}^{({sr})}} \right\}^{r} \right\rbrack + {\sum\limits_{q\rightarrow p}^{q < p}\; {Z_{pq}\left( I_{q} \right)}^{({r + 1})}} + {\sum\limits_{q\rightarrow p}^{q > p}\; {Z_{pq}\left( I_{q} \right)}^{(r)}}} \right\rbrack + {({nk})\left( {ZK}_{p} \right)\left( {IK}_{p} \right)^{(r)}}} & \left( {73d} \right)\end{matrix}$ Full Complex Matrix-Z Formulation: $\begin{matrix}{\mspace{79mu} {V_{p} = {{\sum\limits_{q = 1}^{n}\; {Z_{pq}I_{q}}} = {\sum\limits_{q = 1}^{n}\; {Z_{pq}\left( {{SSH}_{q}^{*}/V_{q}^{*}} \right)}}}}} & \left( {73e} \right) \\\left. {\left. \left. {\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {Z_{pp}\left( {{SSH}_{p}^{*}/V_{p}^{*}} \right)}^{({sr})}} \right\}^{(r)} \right\rbrack + {\sum\limits_{q = 1}^{p - 1}\; {Z_{pq}\left( {{SSH}_{q}^{*}/V_{q}^{*}} \right)}^{(r)}} + {\sum\limits_{q = {p + 1}}^{n}\; {Z_{pq}\left( {{SSH}_{q}^{*}/V_{q}^{*}} \right)}^{(r)}}} \right) & \left( {73f} \right) \\\left. \left. {\left\lbrack V_{p}^{({{sr} + 1})} \right\rbrack^{({r + 1})} = {{Z_{pp}\left( {{SSH}_{p}^{*}/\left\{ \left( V_{p}^{*} \right)^{({sr})} \right\}^{(r)}} \right\rbrack} + \left\{ {\sum\limits_{q = 1}^{p - 1}\; {Z_{pq}\left( {{SSH}_{q}^{*}/V_{q}^{*}} \right)}^{({r + 1})}} \right) + {\sum\limits_{q = {p + 1}}^{n}\; {Z_{pq}\left( {{SSH}_{q}^{*}/V_{q}^{*}} \right)}^{(r)}}}} \right) \right\} & \left( {73g} \right)\end{matrix}$ Sparse Complex Matrix-Z Formulation: $\begin{matrix}{\mspace{79mu} {{\Delta \; V_{p}} = {{Z_{pp}\Delta \; I_{p}} + {\sum\limits_{q->p}\; {Z_{pq}\Delta \; I_{q}}}}}} & \left( {74a} \right) \\{{\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\left( {n - 1} \right)\left( {ZK}_{p} \right)\left( {\Delta \; {IK}_{p}} \right)^{(r)}}}:{{from}\mspace{14mu} \left( {71a} \right)}}},\left( {71b} \right)} & \left( {74b} \right) \\{{{\left. {\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q->p}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}}}} \right) + {({nk})\left( {ZK}_{p} \right)\left( {\Delta \; {IK}_{p}} \right)^{(r)}}}:{{from}\mspace{14mu} \left( {71c} \right)}},\left( {71d} \right)} & \left( {74c} \right) \\{\left. {\left. {\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q\rightarrow p}^{q > p}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{({r + 1})}}}} \right) + {\sum\limits_{q\rightarrow p}^{q > p}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}}} \right) + {({nk})\left( {ZK}_{p} \right)\left( {\Delta \; {IK}_{p}} \right)^{(r)}}} & \left( {74d} \right)\end{matrix}$ Full Complex Matrix-Z Formulation: $\begin{matrix}{\mspace{79mu} {{\Delta \; V_{p}} = {\sum\limits_{q = 1}^{n}\; {Z_{pq}\Delta \; I_{q}}}}} & \left( {74e} \right) \\\left. {\left. {\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q = 1}^{p - 1}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}}}} \right) + {\sum\limits_{q = {p + 1}}^{n}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}}} \right) & \left( {74f} \right) \\\left. {\left. {\left\lbrack {\Delta \; V_{p}^{({{sr} + 1})}} \right\rbrack^{({r + 1})} = {{Z_{pp}\left\lbrack \left\{ \left( {\Delta \; I_{p}^{*}} \right)^{({sr})} \right\}^{(r)} \right\rbrack} + {\sum\limits_{q = 1}^{p - 1}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{({r + 1})}}}} \right) + {\sum\limits_{q = {p + 1}}^{n}\; {Z_{pq}\left( {\Delta \; I_{q}^{*}} \right)}^{(r)}}} \right) & \left( {74g} \right) \\{\mspace{79mu} {{{V_{p}^{({r + 1})} - V_{p}^{(r)}}} \leq ɛ}} & (75)\end{matrix}$ and where, matrix [Z] can also be made-up of real orcomplex components, which is an inverse of coefficient matrix of linearand non-linear equations organized in different possible ways includingin super-decoupled form, or an inverse of the Jacobian [J]⁻¹ or itsdifferent constant or approximated variations including decoupled orsuper decoupled versions, and It should be noted that equations (72a)and (72b) are the same as (60a) and (60) respectively, and (60s) and(63s) are different variations of (60), evaluating loadflow computationfor any over loaded components of the power network and for under orover voltage at any of the nodes of the power network, correcting one ormore controlled variables and repeating the performing loadflowcomputation, evaluating, and correcting steps until evaluating stepfinds no over loaded components and no under or over voltages in thepower network, and affecting a change in power flow through componentsof the power network and voltage magnitudes and angles at the nodes ofthe power network by actually implementing the finally obtained valuesof controlled variables after evaluating step finds a good power systemor stated alternatively the power network without any overloadedcomponents and under or over voltages, which finally obtained controlledvariables however are stored for acting upon fast in case a simulatedevent actually occurs.
 5. A method of forming and solving a model of asystem, a network, an equipment, an apparatus, a device or a material toaffect control of controlled variables/parameters in the system, thenetwork, the equipment, the apparatus, the device or the material,comprising the steps of: obtaining on-line or simulated data of physicalstatus of all compnents of the system, the network, the equipment, theapparatus, the device or the material and their maximum and minimumoperating and physical capability limits, obtaining on-line readings ofspecified/known/given/set variables/parameters, wherein said on-linereadings are the controlled variables/parameters, performing computationby forming and solving a computation model of the system, the network,the equipment, the apparatus, the device or the material to calculatethe unknown variables/parameters, in dependence on the said obtainedon-line readings of specified/known/given/set values of the controlledvariables/parameters and operational and physical limits of thecomponents of the system, the network, the equipment, the apparatus, thedevice or the material, the said computation model of the system, thenetwork, the equipment, the apparatus, the device or the material isreferred to as Patel Computation Model (PCM) characterized by andderived from the following attributes: organizing linear or nonlinearequations as mismatch functions equated to zero, in each of the mismatchfunctions, club any term with known quantities or value into a diagonalterm with simple algebraic manipulations, expressing a vector of themismatch functions as a product of a coefficient matrix and a vector ofunknown variables, which can sometimes be treated as a correction vectorof unknown variables, equating the vector of mismatch functions to theproduct of the coefficient matrix and the vector of unknown variables orthe correction vector of unknown variables to be calculated, solvingsuch a matrix equation by iterations for the vector of unknown variablesor the correction vector of unknown variables using evaluation of thevector of mismatch functions with guess values of unknown variables tobegin with, and inverting or factoring the coefficient matrix,evaluating solution of Patel Computation Model for any violation ofoperational and physical limits of the components of the system, thenetwork, the equipment, the apparatus, the device or the material,correcting one or more controlled variables and repeating the performingcomputation, evaluating, and correcting steps until evaluating stepfinds no violation of operating and physical limits of the components ofthe system, the network, the equipment, the apparatus, the device or thematerial, affecting a change in controlled variables/parameters of thecomponents of the system, the network, the equipment, the apparatus, thedevice or the material by actually implementing the finally obtainedvalues of controlled variables/parameters after evaluating step finds agood or stated alternatively no violations of the operational andphysical limits of the components of the system, the network, theequipment, the apparatus, the device or the material.